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STRUCTURAL  DYNAMICS:  RECENT  ADVANCES 
Proceedings  of  the  6th  International  Conference 
Volume  II 


Proceedings  of  the  Sixth  International  Conference  on  Recent  Advances  in 
Structural  Dynamics,  held  at  the  Institute  of  Sound  and  Vibration  Research, 
University  of  Southampton,  England,  from  14th  to  17th  July,  1997,  co-sponsored  by 
the  US  Airforce  European  Office  of  Aerospace  Research  and  Development  and  the 
Wright  Laboratories,  Wright  Patterson  Air  Force  Base. 


Edited  by 

N,S.  FERGUSON 

Institute  of  Sound  and  Vibration  Research, 
University  of  Southampton,  Southampton,  UK. 

H.F.  WOLFE 

Wright  Laboratory, 

Wright  Patterson  Air  Force  Base,  Ohio,  USA. 
and 
C  MEI 

Department  of  Aerospace  Engineering, 

Old  Dominion  University,  Norfolk,  Virginia,  USA. 


©  The  Institute  of  Sound  and  Vibration  Research,  University  of  Southampton,  UK. 

ISBN  no.  0-85432-6375 


19970814  055 


VV  T.-  ,  .. 


PREFACE 


The  papers  contained  herein  were  presented  at  the  Sixth  International 
Conference  on  Recent  Advances  in  Structural  Dynamics  held  at  the  Institute  of 
Sound  and  Vibration  Research,  University  of  Southampton,  England  in  July  1997. 
The  conference  was  organised  and  sponsored  by  the  Institute  of  Sound  and 
Vibration  Research  and  co-sponsored  by  the  Wright  Laboratories,  Wright  Patterson 
Air  Force  Base.  We  wish  to  also  thank  the  following  for  their  contribution  to  the 
success  of  the  conference:  the  United  States  Air  Force  European  Office  of  Aerospace 
Research  and  Development.  The  conference  follows  equally  successful  conferences 
on  the  same  topic  held  at  Southampton  in  1980, 1984,1988,1991  and  1994. 

There  are  over  one  hundred  papers  written  by  authors  from  approximately 
20  different  countries,  making  it  a  truly  international  forum.  Many  authors  have 
attended  more  than  one  conference  in  the  series  whilst  others  attended  for  the  first 
time. 


It  is  interesting  to  note  the  change  in  emphasis  of  the  topics  covered. 
Analytical  and  numerical  methods  have  featured  strongly  in  all  the  conferences. 
This  time,  system  identification  and  power  flow  techniques  are  covered  by  even 
more  papers  than  previously.  Also,  there  are  many  contributions  in  the  field  of 
passive  and  active  vibration  control.  Papers  dealing  with  nonlinear  aspects  of 
vibration  continue  to  increase.  These  observations  seem  to  reflect  the  trend  in 
current  research  in  structural  dynamics.  We  therefore  hope  that  the  present  series 
of  International  Conferences  will  play  a  part  in  disseminating  knowledge  in  this 
area. 


We  would  like  to  thank  the  authors,  paper  reviewers  and  session  chairmen 
for  the  part  they  played  in  making  it  a  successful  conference. 

My  personal  thanks  go  to  the  following  individuals  who  willingly  and 
enthusiastically  contributed  to  the  organisation  of  the  event: 

Dr.  H.F.  Wolfe  Wright  Laboratories,  WPAFB,  USA 

Dr.  C.  Mei  Old  Dominion  University,  USA 

Mrs.  M.Z.  Strickland  ISVR,  University  of  Southampton,  UK 

Grateful  thanks  are  also  due  to  many  other  members  of  ISVR  who  contributed  to 
the  success  of  the  event. 


N.S.  Ferguson 


Sixth  International  Conference  on 
Recent  Advances  in  Structural  Dynamics 

Volume  II 


Contents 

Page  No. 

INVITED  PAPER 

R.D.  BLEVINS 

On  random  vibration,  probability  and  fatigue  881 

ACOUSTIC  FATIGUE  I 

58.  J.  LEE  and  K.R.  WENTZ 

Strain  power  spectra  of  a  thermally  buckled  plate  in 

random  vibration  903 

59.  S.A.  RIZZI  and  T.L.  TURNER 

Enhanced  capabilities  of  the  NASA  Langley  thermal 

acoustic  fatigue  apparatus  919 

60.  I.  HOLEHOUSE 

Sonic  fatigue  characteristics  of  high  temperature  materials 

and  structures  for  hypersonic  flight  vehicle  applications  935 

61 .  M.  FERMAN  and  H.F.  WOLFE 

Scaling  concepts  in  random  acoustic  fatigue  953 

ACOUSTIC  FATIGUE  II 

62.  H.F.  WOLFE  and  R.G.  WHITE 

The  development  and  evaluation  of  a  new  multimodal 

acoustic  fatigue  damage  model  969 

63.  B.  BENCHEKCHOU  and  R.G.  WHITE 

Acoustic  fatigue  and  damping  technology  in  composite 
materials  981 

64.  D.  MILLAR 

The  behaviour  of  light  weight  honeycomb  sandwich  panels 

under  acoustic  loading  995 

65.  P.D.  GREEN  and  A.  KILLEY 

Time  domain  dynamic  Finite  Element  modelling  in  acoustic 

fatigue  design  1007 


SYSTEM  IDENTIFICATION  II 


66. 

U.  PRELLS,  A.W.  LEES,  M.I.  FRISWELL  and  M.G.  SMART 
Robust  subsystem  estimation  using  ARMA-modelling  in 
the  frequency  domain 

1027 

67. 

Y.Q.  NI,  J.M.  KO  and  C.W.  WONG 

Mathematical  hysteresis  models  and  their  application  to 
nonlinear  isolation  systems 

1043 

68. 

M.G.  SMART,  M.I.  FRISWELL,  A.W.  LEES  and  U.  PRELLS 

The  identification  of  turbogenerator  foundation  models 
from  run-down  data 

1059 

69. 

C.  OZTURK  and  A.  BAHADIR 

Shell  mode  noise  in  reciprocating  refrigeration 
compressors 

1073 

70. 

T.H.T.  CHAN,  S.S.  LAW  and  T.H.  YUNG 

A  comparative  study  of  moving  force  identification 

1083 

71. 

P.A.  ATKINS  and  J.R.  WRIGHT 

Estimating  the  behaviour  of  a  nonlinear  experimental  multi 
degree  of  freedom  system  using  a  force  appropriation 
approach 

1099 

POWER  FLOW  TECHNIQUES  II 

72. 

R.S.  LANGLEY,  N.S.  BARDELL  and  P.M.  LOASBY 

The  optimal  design  of  near-periodic  structures  to  minimise 
noise  and  vibration  transmission 

1113 

73. 

J.L.  HORNER 

Effects  of  geometric  asymmetry  on  vibrational  power 
transmission  in  frameworks 

1129 

74. 

M.  IWANIEC  and  R.  PANUSZKA 

The  influence  of  the  dissipation  layer  on  energy  flow  in 
plate  connections 

1143 

75. 

H.  DU  and  F.F.  YAP 

Variation  analysis  on  coupling  loss  factor  due  to  the  third 
coupled  subsystem  in  Statistical  Energy  Analysis 

1151 

76. 

S.J.  WALSH  and  R.G.  WHITE 

The  effect  of  curvature  upon  vibrational  power 
transmission  in  beams 

1163 

77. 

S.  CHOI,  M.P.  CASTANIER  and  C.  PIERRE 

A  parameter-based  statistical  energy  method  for  mid¬ 
frequency  vibration  transmission  analysis 

1179 

PASSIVE  AND  ACTIVE  CONTROL  III 


78.  Y.  LEI  and  L.  CHEN 

Research  on  control  law  of  active  suspension  of  seven 

degree  of  freedom  vehicle  model  1195 

79.  M.  AHMADIAN 

Designing  heavy  truck  suspensions  for  reduced  road 
damage  1203 

80.  A.M.  SADRI,  J.R.  WRIGHT  and  A.S.  CHERRY 

Active  vibration  control  of  isotropic  plates  using 
piezoelectric  actuators  1217 

81.  S.M.  KIM  and  M.J.  BRENNAN 

Active  control  of  sound  transmission  into  a  rectangular 
enclosure  using  both  structural  and  acoustic  actuators  1233 

82.  T.J.  SUTTON,  M.E.  JOHNSON  and  S.J.  ELLIOTT 

A  distributed  actuator  for  the  active  control  of  sound 
transmission  through  a  partition  1247 

83.  J.  RO,  A.  A-ALI  and  A.  BAZ 

Control  of  sound  radiation  from  a  fluid-loaded  plate  using 

active  constraining  layer  damping  1257 


ANALYTICAL  METHODS  11 

84.  E.  MANOACH,  G.  DE  PAZ,  K.  KOSTADINOV  and 
F.  MONTOYA 

Dynamic  response  of  single-link  flexible  manipulators  1275 

85.  B.  KANG  and  C.A.  TAN 

Wave  reflection  and  transmission  in  an  axially  strained, 
rotating  Timoshenko  shaft  1291 

86.  Y.  YAMAN 

Analytical  modelling  of  coupled  vibrations  of  elastically 
supported  channels  1329 

87.  R.S.  LANGLEY 

The  response  of  two-dimensional  periodic  structures  to 
harmonic  and  impulsive  point  loading  1345 

NONLINEAR  VIBRATION  III 

88.  H.  OYANG,  J.E.  MOTTERSHEAD,  M.P.  CARTMELL  and 
M.L  FRISWELL 

Stick-slip  motion  of  an  elastic  slider  system  on  a  vibrating 
disc 


1359 


89. 

R.Y.Y.  LEE,  Y.  SHI  and  C.  MEI 

A  Finite  Element  time  domain  multi-mode  method  for 
large  amplitude  free  vibration  of  composite  plates 

1375 

90. 

P.  RIBEIRO  and  M.  PETYT 

Nonlinear  forced  vibration  of  beams  by  the  hierarchical 
Finite  Element  method 

1393 

91. 

K.M.  HSIAO  and  W.Y.  LIN 

Geometrically  nonlinear  dynamic  analysis  of  3-D  beam 

1409 

92. 

R.Y.Y.  LEE,  Y.  SHI  and  C.  MEI 

Nonlinear  response  of  composite  plates  to  harmonic 
excitation  using  the  Finite  Element  time  domain  modal 
method 

1423 

93. 

C.W.S.  TO  and  B.  WANG 

Geometrically  nonlinear  response  analysis  of  laminated 
composite  plates  and  shells 

1437 

ANALYTICAL  METHODS  III 

94. 

R.S.  HWANG,  C.H.J.  FOX  and  S.  McWILLIAM 

The  free,  in-plane  vibration  of  circular  rings  with  small 
thickness  variations 

1457 

95. 

D.J.  GORMAN 

Free  vibration  analysis  of  transverse-shear  deformable 
rectangular  plates  resting  on  uniform  lateral  elastic  edge 
support 

1471 

96. 

R.G.  PARKER  and  C.D.  MOTE,  Jr. 

Wave  equation  eigensolutions  on  asymmetric  domains 

1485 

97. 

A.V.  PESTEREV 

Substructuring  for  symmetric  systems 

1501 

RANDOM  VIBRATION  I 

98. 

G.FUandJ.  PENG 

Anaytical  approach  for  elastically  supported  cantilever 
beam  subjected  to  modulated  filtered  white  noise 

1517 

99. 

S.D.  FASSOIS  and  K.  DENOYER 

Linear  multi-stage  synthesis  of  random  vibration  signals 
from  partial  covariance  information 

1533 

100. 

CW.S.TOand  Z.  CHEN 

First  passage  time  of  multi-degrees  of  freedom  nonlinear 
systems  under  narrow-band  non-stationary  random 
excitations 

1549 

101. 

C.  FLORIS  and  M.C.  SANDRELLI 

Random  response  of  Duffing  oscillator  excited  by  quadratic 
polynomial  of  filtered  Gaussian  noise 

1565 

102. 

S.  McWILUAM 

Extreme  response  analysis  of  non-linear  systems  to  random 
vibration 

1581 

103. 

M.  GHANBARI  and  J.F.  DUNNE 

On  the  use  of  Finite  Element  solutions  of  the  FPK  equation 
for  non-linear  stochastic  oscillator  response 

1597 

RANDOM  VIBRATION  II 

104. 

T.L.  PAEZ,  S.  TUCKER  and  C.  O’GORMAN 

Simulation  of  nonlinear  random  vibrations  using  artificial 
neural  networks 

1613 

105. 

D.Z.  LI  and  Z.C.  FENG 

Dynamic  properties  of  pseudoelastic  shape  memory  alloys 

1629 

106. 

Z.W.  ZHONG  and  C.  MEI 

Investigation  of  the  reduction  in  thermal  deflection  and 
random  response  of  composite  plates  at  elevated 
temperatures  using  shape  memory  alloys 

1641 

SIGNAL  PROCESSING  I 

107. 

M.  FELDMAN  and  S.  BRAUN 

Description  of  non-linear  conservative  SDOF  systems 

1657 

108. 

N.E.  KING  and  K.  WORDEN 

A  rational  polynomial  technique  for  calculating  Hilbert 
transforms 

1669 

109. 

D.M.  LOPES,  J.K.  FIAMMOND  and  P.R.  WHITE 

Fractional  Fourier  transforms  and  their  interpretation 

1685 

SYSTEM  IDENTIFICATION  III 

no.  J.  DICKEY,  G.  MAIDANIK  and  J.M.  D’ARCHANGELO 

Wave  localization  effects  in  dynamic  systems  1701 

111.  P.  YUAN,  Z.F-  WU  and  X.R.  MA 

Estimated  mass  and  stiffness  matrices  of  shear  building 
from  modal  test  data 


1713 


112.  YU.  I.  BOBROVNirSKn 

The  problem  of  expanding  the  vibration  field  from  the 
measurement  surface  to  the  body  of  an  elastic  structure  1719 

113  M.  AMABILI  and  A.  FREGOLENT 

Evaluation  of  the  equivalent  gear  error  by  vibrations  of  a 
spur  gear  pair 


1733 


ON  RANDOM  VIBRATION,  PROBABILITY,  AND  FATIGUE 


R.  D.  Blevins 

Rohr  Inc.,  Mail  Stop  107X 

850  Lagoon  Drive 

Chula  Vista,  California  91910 


ABSTRACT 

Analysis  is  made  to  determine  the  properties  of  a  random  process  consisting  of  the 
sum  of  a  series  of  sine  waves  with  deterministic  amplitudes  and  independent,  random 
phase  angles.  The  probability  density  of  the  series  and  its  peaks  are  found  for  an  arbitrary 
number  of  terms.  These  probability  distributions  are  non-Gaussian.  The  fatigue  resulting 
from  the  random  vibration  is  found  as  a  function  of  the  peak-to-rms  ratio. 


1.  INTRODUCTION 

Vibration  spectra  of  aircraft  components  often  are  dominated  by  a  relatively  small 
number  of  nearly  sinusoidal  peaks  as  shown  in  Figure  1.  The  time  history  of  this  process, 
shown  in  Figure  2,  is  irregular  but  bounded.  The  probability  density  of  the  time  history, 
shown  in  figure  3  only  roughly  approximates  a  Gaussian  distribution  and  it  does  not  exceed 
2.5  standeird  deviations. 

The  time  history  of  displacement  or  stress  of  these  processes  over  a  flight  or  a  take 
off  time  can  be  expressed  as  a  Fourier  series  of  a  finite  number  of  terms  over  the  finite 
sampling  period  T. 

N 

y  =  0,nCOs(u)-ntn  +  <^n),  0  <  in  <  T,  Un  >  0  (l) 

n=l 

Each  frequency  Un  is  a  positive,  non-zero  integer  multiple  of  27r/r.  The  following  model  is 
used  for  the  nature  of  the  Fourier  series:  1)  the  amplitudes  a-n  are  positive  and  deterministic 
in  the  sense  that  they  do  not  vary  much  from  sample  to  sample,  2)  the  phases  (j>n  are  random 
in  the  sense  that  they  vary  from  sample  to  sample,  they  are  equally  likely  to  occur  over 
the  range  -oo  <  0n  <  oo.  This  last  condition  implies  that  the  terms  on  the  right  hand 
side  of  equation  (1)  are  statistically  independent  of  each  other. 

We  can  generate  an  ensemble  of  values  of  the  dependent  variable  Y  by  randomly 
choosing  M  sets  of  N  phase  angles  =  1,2..^),  computing  Y  at  some  ffxed  time 

from  equation  (1),  choosing  another  set  of  phases,  computing  a  second  value  of  Y  and  so 


881 


on  until  we  have  a  statistically  significant  sample  of  M  Y's.  This  random  phase  approach, 
introduced  by  Rayleigh  (1880),  models  a  multi-frequency  processes  where  each  frequency 
component  is  independent  and  whose  power  spectral  density  (PSD)  is  known. 

The  maximum  possible  (peak)  value  of  equation  (1)  is  the  sum  of  the  amplitude  of 
each  term  (recall  >  0).  The  mean  square  of  the  sum  of  independent  sine  waves  is  the 
sum  of  the  mean  squares  of  the  terms. 


N 


^peak  ~  'y  ^ 


n=l 


=  Na,  for  ai=  02=  an  =  CL 


(2a) 

(2&) 


X  N  N 

Yrms  =  ^  f  [Y  anCO«(2wt„/T  +  <Pri)?dtn  =  5 

=  |iVa^,  for  ai  =  02  =  an  =  a 

The  peak-to-rms  ratio  of  the  sum  of  N  mutually  independent  sine  waves  thus  is, 

N  N 


I  rms 


1/2 


n=l  71=1 

=  (2A/')^/^,  for  Oi  =  02  =  an  =  o. 


(3a) 

{Sb) 

(4a) 

(46) 


Equation  (4b)  shows  that  the  peak-to-rms  ratio  for  an  equal  amplitude  series  increases  from 
2^/^  for  a  single  term  (N=l)  and  approaches  infinity  as  the  number  of  terms  N  approaches 
infinity,  as  shown  in  Figure  4.  The  probability  of  Y  is  zero  beyond  the  peak  value.  For 
example,  there  is  no  chance  that  the  sum  of  any  four  {N  =  4)  independent  sinusoidal  terms 
will  be  greater  than  8^/^  =  2.828  times  the  overall  rms  value. 


2.  PROBABILITY  DENSITY  OF  A  SINE  WAVE 

The  probability  density  py  (y)  of  the  random  variable  Y  is  probability  that  the  random 
variable  Y  has  values  within  the  small  range  between  y  and  y  -b  dy,  divided  by  dy.  p(Y) 
has  the  units  of  1/Y.  Consider  single  a  sine  wave  of  amplitude  a^,  circular  frequency  uJri) 
and  phcLse 

Y  =  On  COs{0Jntn  +  0  <  <  277.  (5) 

Y  is  the  dependent  random  variable.  The  independent  random  variables  are  tn  or  <l>n-  The 
probability  density  of  a  sine  wave  for  equal  likely  phases  p((l>n)  =  l/(27r),  or  equally  likely 
times,  p[tn)  =  1/T,  is  (Bennett,  1944;  Rice  1944,  art.  3.10), 


wiy)  = 


77  ^(a^  -  j/2)  1/2^  if  <y  <an\ 


0, 


if  I2/I  >  CLn 


(6) 


882 


The  probability  density  of  the  sine  wave  is  symmetric  about  y  =  0,  i.e.,  pyiv)  ~  PYi~y)) 
it  is  singular  at  y  =  Cn,  and  it  falls  to  zero  for  jyl  greater  than  an  as  shown  in  Figure  5. 

The  characteristic  function  of  a  random  variable  x  is  the  expected  value  of 

C{f)  =  r  (7) 

J  —CO 

and  it  is  also  the  Fourier  transform  of  the  probability  density  function  (Cramer,  1970, 
pp.  24-35;  Sveshnikov,  1965;  with  notation  of  Bendat,  1958).  j  =  is  the  imaginary 
constant.  The  characteristic  function  of  the  sine  wave  is  found  using  equations  (13)  and 
(14)  and  integrating  over  the  range  0  <  X  <  a^.  (Gradshteyn,  Ryzhik,  Jeffrey,  1994,  article 
3.753). 


Cn(f)  =  2(7ra„)-‘  T"  cos{27rfY))[l  -  {Y / dY  =  Jo(27r/a„),  (8) 

Jo 

The  characteristic  function  of  a  sine  wave  is  a  Bessel  function  of  the  first  kind  and  zero 
order  (Rice,  1944,  art.  3.16).  Equations  (6)  and  (8)  are  starting  points  for  determining 
the  probability  density  of  the  Fourier  series. 


3.  PROBABILITY  DENSITY  OF  THE  SUM  OF  N  SINE  WAVES 

It  is  possible  to  generate  an  expression  for  the  probability  density  of  Fourier  series 
(equation  l)  with  1,2, 3, to  any  number  of  terms  provided  the  sine  wave  terms  are  mutually 
independent.  This  is  done  with  characteristic  functions.  The  characteristic  function  of  the 
sum  of  N  mutually  independent  random  variables  (Y  =  Xi  -j-  X2  +  -■■i-  Xj\/)  is  the  product 
of  their  characteristic  functions  (Weiss,  1990,  p.22;  Sveshnikov,  pp.  124-129), 

c(/)  =  r  ..  r  e^2-/{^.+^=+- 

7  —00  7—00 

N  ^00  N 

=  n  /  =  n  CM)-  (9) 

The  symbol  11  denotes  product  of  terms.  The  characteristic  function  for  the  sum  of  N 
independent  sine  waves  is  found  from  equations  (8)  and  (9). 


C{f)^ 


n!Li  *^0 (27r/an) ,  unequal  an 
[Jo(27r/a)]^,  ai  ==  a2  =  =  a 


(10) 


The  probability  density  of  Y  is  the  inverse  Fourier  transform  of  its  characteristic  function 
(Sveshnikov,  1968,  p.  129). 


py{y)  =  r  e-^^-fyc{f)df 

7—00 


(11) 


883 


By  substituting  equation  (10)  into  equation  (11)  we  obtain  an  integral  equation  for  the 
probability  density  of  a  N-term  finite  Fourier  series  of  independent  sine  waves  (Barakat, 
1974). 


.oo  N 

Pviy)  =  2  /  cos{2Tryf)  {  TT  Jo(27r/an)  }  df, 

''0  n=l 


iV  =  1,2,3... 


(12) 


If  all  N  terms  of  the  Fourier  series  have  equal  amplitudes  a  =  ai  —  a2  =  On  =  a/sr,  then 
this  simplifies, 


py(y)  =  2  r  cos(2iryf)[Jo{2wfa)fdf,  N  =  1,2,3...  (13) 

Jo 

These  distributions  are  symmetric  about  y  —  0  as  are  all  zero  mean,  sum-of-sine-wave 
distributions.  Figures  5  and  6  show  results  of  numerically  integrating  equations  (45)  and 
(46)  over  interval  /  =  0  to  /  =  15a  using  Mathematica  (Wolfram,  1995). 

Barakat  (1974,  also  see  Weiss,  1994,  p.  25)  found  a  Fourier  series  solution  to  equation 
(45).  He  expanded  the  probability  density  of  the  N  term  sum  in  a  Foui'ier  series  over  the 
finite  interval  -Ly  <  Y  <  Ly  where  Ly  =  ai  +  a2  +  -  +  The  result  for  unequal 
amplitudes  is  , 


\v\<Ly. 

i=L  n=l 

For  equal  amplitudes,  ai  =  a2  =  a,^  =  o,  Ly  =  Na,  and 


i=l 


\y\  <  Na. 


(14) 


(15) 


Figure  6  shows  that  the  Fourier  series  solution  (equation  15)  carried  to  20  terms  to  be 
virtually  identical  to  numerical  integration  of  equation  (13)  and  it  compares  well  with  the 
approximate  solution.  Note  that  theory  requires  py{\yT\  >  Ly)  =  0. 

A  power  series  solution  for  equation  (13)  can  be  found  with  a  technique  used  by  Rice 
(1944,  art.  16)  for  shot  noise  and  by  Cramer  (1970)  who  called  it  an  Edgeworth  series.  The 
Bessel  function  term  in  equation  (13)  is  expressed  as  an  exponent  of  a  logarithm  which  is 
then  expanded  in  a  power  series, 

[Jo(27r/a)]-^  =  ex'p{N  ln[Jo[2'Kaf)\),  (16) 

=  exvi-Nir'^a^f  -  (l/4)iV7r^o'‘/''  -  (l/9)N-K^a^f  +  (n/192)Arx5a®/-)- 

=  +  -1 

Substituting  this  expansion  into  equation  (13)  and  rearranging  gives  a  series  of  integrals, 
which  are  then  solved  (Gradshteyn,  Ryzhik,  Jeffrey,  1994,  arts.  3.896,  3.952)  to  give  a 


884 


power  series  for  the  probability  density  of  the  equal-amplitude  N-term  Fourier  series  sum. 


VY{y)  = 


g^iFi[-2,V2,yV(2yA.,)l 


-( 


11 


r(9/2) 


(17) 


192iV3  32iv2^  ^1/2 


)^^iFi[-4,l/2,yV(2i;L.)l  +  -)^  \y\<Na 


'PY{\y\  >  =  0  and  Yrms  is  given  by  equation  (3b).  There  are  two  special  func¬ 

tions  in  equation  (17),  the  gamma  function  T  and  the  confluent  hypergeometric  function 
iFi[n,'y,z].  These  are  defined  by  Gradshteyn,  Ryzhik,  and  Jeffrey  (1994). 


As  N  approaches  infinity,  the  peak-to-rms  (equation  4b)  ratio  approaches  infinity,  and 
equation  (51)  approaches  the  normal  distribution, 

\im  pY{y)  =  ~;^ — .  (18) 

N-*oo  V^Yrms 

as  predicted  by  the  central  limit  theorem  (Cramer,  1970;  Lin,  1976). 


4.  PROBABILITY  DENSITY  OF  PEAKS 

Theories  for  calculating  the  fatigue  damage  from  a  time  history  process  generally 
require  knowledge  of  the  peaks  and  troughs  in  the  time  history.  This  task  is  made  simpler 
if  we  assume  that  the  time  history  is  narrow  band.  If  Y{t)  is  narrow  band  that  is,  that 
each  trajectory  of  Y{t)  which  crosses  zero  has  only  a  single  peak  before  crossing  the  cixis 
again,  then  (1)  the  number  of  peaks  equals  the  number  of  times  the  time  history  crosses 
the  axis  with  positive  slope,  and  (2)  only  positive  peaks  occur  for  Y{t)  >0  and  they  are 
located  at  points  of  zero  slope,  dY{t)ldt  =  0.  Lin  (1967,  p.  304)  gives  expressions  for  the 
expected  number  of  zero  crossings  with  positive  slope  (peaks  above  the  axis)  per  unit  time 
for  a  general,  not  necessarily  narrow  band,  process, 

ElNo+]=  f  ypyy(0,y)<iy  (19) 

Jo 

and  the  probability  density  of  the  peaks  for  a  narrow  band  process. 

=  (20) 

In  order  to  apply  these  expressions,  the  joint  probability  distribution  of  Y  and  Y  must  be 
established.  The  joint  probability  density  function  Pyriy^y)  random  variable 

Y  and  Y  is  the  probability  that  Y  falls  in  the  range  between  y  and  y  +  dy  and  y  falls  in  the 
range  between  y  and  y-\-dy,  divided  by  dydy.  The  derivative  of  the  sine  wave  Y  (equation 
12)  with  respect  to  time  can  be  expressed  in  terms  of  Y , 

dY/dt  =  Y  —  -Gn^n  sin{u)nt  +  0n)  ==  ~  Y^,  \Y\  <  On.  (21) 


885 


The  joint  probability  density  is  the  inverse  Fourier  transform  of  its  characteristic 
function, 

/CO  poo 

/  (30) 

-oo  J—oo 

The  proof  of  equations  (28),  (29),  and  (30)  can  be  found  in  Chandrasekhar  (1943),  Willie 
(1987),  Weiss  and  Shmueli  (1987),  and  Weiss  (1994,  pp.  21-26). 

Since  the  probability  is  symmetric  about  y  —  y  =  0,  Pyriv^v)  ~  Pyy(~S/j“y))  only 
symmetric  terms  survive  the  integration.  Substituting,  equation  (29)  into  equation  (30) 
and  expanding  gives  and  integral  expression  for  the  joint  probability  of  y  and  Y. 

Pyy(2/,y)==  [  [  {JJ  ^o(27ranY^/f  +  /|cj2)}cos(27r/iy)cos(27r/2y)d/i<i/2  (31) 


It  is  also  possible  to  expand  the  joint  probability  of  Y  and  Y  in  as  double  finite  Fourier 
series.  The  result  is: 


..  oo  oo  N  I  .  7 

Pyriv^y)  =  (^)2j}cos(i7ry/Ly)cos(/:7ry/L^) 


aik  =  1,  i,  /c>  0;  1/2,  i  —  Qork  =  0;  1/4,  k  =  0 


(32) 

(33) 


The  expected  number  of  peaks  per  unit  time  and  the  probability  distribution  of  the  narrow 
band  peaks  is  obtained  by  substituting  this  equation  into  equations  (19)  and  (20)  and 
integrating.  The  results  are: 


n—1 


^0  k=0 


TTUr 


Ly  ‘ 


(34) 


y  OO  OO  iV 


:/-)'  + (^P)}sin(i^A/Ly) 
JuY  J^Y 


where 


(  1/8,  i  =  /c  =  0, 

_  J  1/4,  i  >  0,  A:  =  0 

-  1  (l/2)[(-l)'‘-  -  ll/(fc7r)^  i  =  0,fc  >  0, 

I  [(-1)*  -  ll/(fe7r)2,  i>0.fc>0. 


(35) 


(36) 


If  the  frequencies  are  closely  spaced  so  uJn^<^  and  hence  Ly  »  uLy ,  then  one  positive 
peak  is  expected  once  per  cycle, 


£;[iVo+]=a;/(27r) 


(37) 


886 


and  the  probability  density  of  narrow  band  peaks  becomes, 


N 


t=0  fc=0 


71=1 


(38) 


Figure  7  shows  probability  density  of  narrow  band  peaks  for  N=2,3,  and  4  equal  amplitude 
(tti  =  1)  equal  frequency  series  using  equation  (38).  Each  sum  in  equation  (38)  was  carried 
to  40  terms. 

A  power  series  solution  for  equation  (20)  can  be  found  if  all  N  terms  in  the  series  have 
equal  amplitude  and  frequency.  The  result  is 


Pa(A)  = 


Y2 


1  1 
4iV  4ivy2 


A^ 


32Nr,i, 


•h... 


In  the  limit  as  N  becomes  infinite  these  equations  become, 


(39) 


pYi-(y>y)  = 


IttY  Y 

Zr/i  j  rms*^  rms 


(40) 


Pa{A)  = 


y2 

^  rms 


(41) 


Equation  (40)  is  in  agreement  with  an  expression  given  by  Crandall  and  Mark  (1963,  p. 
47)  and  equation  (41) is  the  Rayleigh  distribution. 


Equations  (20),  (35),  (38),  and  (41)  are  conservative  when  applied  to  non-narrow  band 
processes  in  the  sense  that  any  troughs  above  the  axis  (points  with  y  >  0  and  dYjdt  =  0 
but  d^Yldt^  >  0)  are  counted  as  peaks  (Lin,  1967,  p.  304;  Powell,  1958;  Broch,  1963). 


Equations  (35),  (38)  and  (41)  can  provide  probability  distributions  for  peaks  of  narrow 
band  processes  as  a  function  of  the  number  of  sine  waves  from  one  to  infinity  and  thus 
they  model  random  processes  with  peak-to-rms  ratios  from  2^/^  to  infinity. 


5.  FATIGUE  UNDER  RANDOM  LOADING 

Fatigue  tests  are  most  often  made  with  constant-amplitude  sinusoidal  loading.  The 
number  of  cycles  to  failure  is  plotted  versus  the  stress  that  produced  failure  and  the  data 
is  often  fitted  with  an  empirical  expression.  MIL-HDBK-5G  (1994)  uses  the  following 
empirical  expression  to  fit  fatigue  data, 

log  iVy  =  +  B2log{S^  -  S4),  =  5(1  -  R)^K  (42) 

Here  Nf  is  the  cycles  to  failure  during  sinusoidal  loading  that  has  maximum  stress  5  per 
cycle.  R  is  the  ratio  of  maximum  to  minimum  stress  during  a  cycle.  R  =  — 1  is  fully 


887 


reversed  stress  cycling.  Bi  though  B4  are  fitted  parameters.  With  a  little  work,  we  can 
put  this  expression  in  the  form  used  by  Crandall  and  Mark  (1963,  p.  113). 

JV  =  cSJ*-  (43) 

where  Sd  =  5(1  -  -  B4,  c  =  10-®' ,  and  b  =  -82-  For  cycling  in  a  time  history  that 

has  non  constant  amplitude,  Miner-Palmgren  proposed  that  the  accumulated  damage  is 
the  sum  of  the  ratios  of  the  number  of  cycles  at  each  amplitude  to  the  allowing  number  of 
cycles  to  failure  at  that  amplitude  (equations  42  and  43). 

D  =  ^«(Si)/lV^(S,)  (44) 

i 

where  n{Si)  is  the  number  of  cycle  accumulated  at  stress  amplitude  Si  and  Nj  is  the 
number  of  stress  cycles  at  this  amplitude  which  would  cause  failure. 

Following  Miles(1954)  and  Crandall  and  Mark(1963),  the  expected  fractional  damage 
for  a  random  stress  cycling  in  system  with  dominant  cycling  at  frequency  f  in  time  t/,  is 

=  (45) 

where  Pa{S)  is  the  probability  density  of  a  stress  cycle  having  amplitude  S  and  Nf{S)  is 
the  number  of  allowable  cycles  to  failure  at  this  stress.  Failure  under  random  loading  is 
expected  when  the  expected  damage  is  unity.  Setting  =  1  at  time  such  that 

ftd  =  Nd,  the  inverse  of  the  expected  number  of  random  vibration  cycles  to  failure  is 

This  expression  can  be  used  to  create  a  fatigue  curve  for  random  cycling  given  the  proba¬ 
bility  density  of  the  random  stress  cycle  amplitudes  (p^(5))  and  a  fatigue  curve  (equation 
42  with  parameters  Bi  though  B4  and  R)  for  sinusoidal  cycling. 

Substituting  the  probability  density  expression  for  narrow  band  amplitude  (equation 
35)  and  for  the  fatigue  curve(equations  42  or  43)  into  equation  (46)  and  integrating,  we 
obtain  an  expression  for  the  expected  number  of  cycles  to  failure  as  a  function  of  the 
number  of  sine  waves  and  their  amplitudes.  For  N  equal  amplitude  sine  waves  this  is, 


<"->■■  -  I  ^1" 


(47) 


V  (2  +  6)L,(l-B)®=  ' 

+ 


iirBi 


(1  -  R)^Ly 
mBi 


■2’ 

.1  3 


1  T-irr-^  f-**  ^ 

(1  - 


iV(B^(l-B)®’-B4)2, 
ALlil  -  B)2B3  ' 

i^7r^(£„(l  -  B)®°  -  Bif. 
4L|(1  -  H)2®» 


888 


Recall  that  for  this  case  Ly  =  Na,  the  rms  value  is  Y^ms  —  {l/2)Na  and  the  peak- 
to-rms  ratio  is  Peak/Yrms  =  y/2N  (equations  2  though  4).  is  the  generalized 

hypergeometric  function  which  is  a  series  of  polynomials.  It  is  described  by  Gradshteyn, 
Ryzhik,  and  Jeffrey  (1994). 

It  is  also  possible  to  establish  the  fatigue  curve  using  the  Rayleigh  distribution  (equa¬ 
tion  18)  and  the  MIL-HDBK-5  fatigue  curve  (equation  42).  The  result  is 


(2^'^r[l  +  5]((1  - 


-BlF,[l  + 


^  3  Bl 

2’2’2y;2„,(i-ij)2B3« 


(48) 


Bl 


rr2Y,^^{i-RYB,^ 


_orf3  +  il  +  3  Bl 

4  2’  2’  2V;2„,(1  -  ii)2B3 


iFi[..]  is  the  confluent  hypergeometric  function  which  is  described  by  Gradshteyn,  Ryzhik, 
and  Jeffrey  (1994). 

Much  of  the  complexity  of  these  last  two  equations  arises  from  the  term  B4  which  is 
associated  with  an  endurance  limit  in  the  fatigue  equation.  That  is,  equation  (42) predicts 
that  sinusoidal  stress  cycling  with  stress  less  than  54/(1— R) ^=3  produces  no  fatigue  damage. 
If  we  set  B4  =  0  to  set  the  endurance  limit  to  zero,  then  equation  (48)  simplifies  to. 

This  result  for  cycles  to  failure  under  Gaussian  loading  without  an  endurance  limit  is  also 
given  by  Crandall  and  Mark  (1963,  p.  117). 

Equations  (47),  (48)  and  (49)  allow  us  to  compute  the  fatigue  curves  of  a  material 
under  random  loading  from  a  fatigue  curve  generated  under  sinusoidal  loading  (equation 
43)  for  narrow  band  random  processes  with  any  peak-to-rms  ratio  from  2^/^  to  infinity. 


6.  APPLICATION 

Figure  8  is  the  MIL-HDBK-5G  fatigue  curve  for  aluminum  2024-T3  with  a  notch 
factor  of  Kt=4  under  sinusoidal  loading  with  various  R  values.  The  fitted  curve  shown  in 
the  figure,  gives  the  following  parameters  for  equation  (42). 

Bl  =  8.3,  B2  =  -3.30,  Bz  =  0.66,  B4  =  8.4 


889 


The  B2  and  B3  are  dimensionless.  B4  has  the  units  of  ksi,  that  is  thousands  of  psi,  and 
10^^  has  units  of  These  Bi,..B4  are  substituted  into  equations  (43),  (47),  (48), 

and  (49). 

The  fatigue  curves  under  random  loading  are  computed  as  follows,  1)  the  number  of 
sine  waves  N  is  chosen  and  this  fixes  the  peak-to-rms  ratio  from  equation  (3b),  2)  set  of 
values  of  rms  stresses  are  chosen  and  for  each  the  corresponding  sine  waves  amplitudes  are 
computed  using  equation  (3b),  a  =  Srmsy/VN  (note  that  the  peak  stress  much  exceed 
S4=8.5  ksi),  and  3)the  cycles  to  failure  are  calculated  from  equation  (47)  for  finite  peak- 
to-rms  ratios  and  equation  (48)  for  Gaussian  loading  (infinite  peak-to-rms). 

For  single  sine  wave,  the  peak-to-rms  ratio  is  2^/^,  equation  4b,  and  the  fatigue  curve 
interms  of  rms  stress  is  adapted  from  the  empirical  data  fit  (equations  42,  43)  by  substi¬ 
tuting  2^^‘^SrTns  for  the  stress  amplitude. 


Nd  =  c(2^/25.n..(l  -  -  B^r^  (50) 

where  b=  -B2  and  c  =  10-®^  Some  results  are  shown  in  Figure  9  for  R=-l. 

7.  CONCLUSIONS 

Analysis  has  been  made  to  determine  the  properties  of  a  random  process  consisting  of 
the  sum  of  a  series  of  sine  waves  with  deterministic  amplitudes  and  random  phase  angles. 
The  joint  probability  density  of  the  sum  and  its  first  two  derivatives  is  determined.  The 
probability  density  of  the  sum  and  narrow  band  peaks  have  been  found  for  an  arbitrary 
number  of  statistically  independent  sine  wave  terms.  The  fatigue  cycles-to-failure  resulting 
from  these  processes  has  been  found. 

1.  The  peak-to-rms  ratio  of  the  sum  of  mutually  independent  terms  exceeds  unity.  If  ail 
terms  have  the  same  peak  and  rms  values  then  the  peak-to-rms  ratio  of  the  series  sum 
increases  with  the  square  root  of  the  number  of  terms  in  the  series.  The  probability 
of  the  series  sum  is  zero  beyond  a  maximum  value,  equal  to  the  sum  of  the  series 
amplitudes,  and  below  the  minimum  value.  Hence,  he  probability  densities  of  the 
finite  series,  their  peaks,  and  their  envelope  are  non  Gaussian. 

3.  The  formulas  allow  the  direct  calculation  of  the  probability  density  of  the  series  and  its 
peaks  from  its  power  spectra  density  (PSD)  under  the  assumption  that  each  spectral 
component  is  statistically  independent. 

4.  The  fatigue  curves  of  a  material  under  random  loading  with  any  peak-to-rms  ratio 
from  2^/^  to  infinity  can  be  computed  dfrectly  from  the  fatigue  curve  of  the  material 
under  sinusoidal  loading. 


890 


REFERENCES 


Abramowitz,  M.  and  LA.  Stegun  1964  Handbook  of  Mathematical  Functions,  National 
Bureau  of  Standards,  U.S.  Government  Printing  OfRce,  Washington  D.C.  Reprinted  by 
Dover. 

Bennett,  W.R.,  1944  Acoustical  Society  of  America  15,  165.  Response  of  a  Linear  Rectifier 
to  Signal  and  Noise. 

Bendat,  J.S.,  1958  Principles  and  Applications  of  Random  Noise  Theory,  Wiley,  N.Y. 

Chandrasekhar,  S.,  1943,Reweiys  of  Modem  Physics,  15,  2-74.  Also  available  in  Wax,  N. 
(ed)  Selected  Papers  on  Noise  and  Stochastic  Processes,  Dover,  N.Y.,  1954. 

Cramer,  H.,  1970  Random  Variables  and  Probability  Distributions,  Cambridge  at  the  Uni¬ 
versity  Press. 

Crandall,  S.H.,  and  C.  H.  Mark  1963  Random  Vibrations  in  Mechanical  Systems,  Academic 
Press,  N.Y. 

Department  of  Defense,  1994  Metallic  Materials  and  Elements  for  Aerospace  Vehicle  Struc¬ 
tures,  MIL-HDBK-5G. 

Gradshteyn,  I.S.,  I.M.  Ryzhik,  and  A.  Jeffrey  1994  Table  of  Integrals,  Series,  and  Products 
5th  Ed.,  Academic  Press,  Boston. 

Lin,  P.K.,  1976  Probabilistic  Theory  of  Structural  Dynamics,  Krieger,  reprint  of  1967  edi¬ 
tion  with  corrections. 

Mathematica,  1995  Ver  2.2,  Wolfram  Research,  Champaign,  Illiinois. 

Miles,  J.,  1954  Journal  of  Aeronautical  Sciences  21,  753-762.  On  Structural  Fatigue  under 
Random  Loading. 

Powell,  A.,  1958  Journal  of  the  Acoustical  Society  of  America  SO  No.  12,  1130-1135.  On 
the  Fatigue  Failure  of  Structure  due  to  Vibrations  Excited  by  Random  Pressure  Fields. 

Rayleigh,  J.W.S.  1880  Philosophical  Magazine  X  73-78.  On  the  Resultant  of  a  Large 
Number  of  Vibrations  of  the  Same  Pitch  and  Arbitrary  Phase.  Also  see  Theory  of  Sound, 
Vol  10,  art.  42a,  reprinted  1945  by  Dover,  N.Y..  and  Scientific  Papers,  Dover,  N.Y.,  1964, 
Vol.  I,  pp.  491-496. 

Rice,  S.O.,  1944  The  Bell  System  Technical  Journal  23  282-332.  Continued  in  1945  24  , 
46-156.  Mathematical  Analysis  of  Random  Noise.  Also  available  in  Wax,  N.  (ed)  Selected 
Papers  on  Noise  and  Stochastic  Processes,  Dover,  N.Y.,  1954. 

Shmulei,  U.  and  G.H.  Weiss  1990  Journal  of  the  American  Statistical  Association  85  6-19. 
Probabilistic  Methods  in  Crystal  Structure  Analysis. 

Sveshnikov,  A. A,  1968  Problems  in  Probability  Theory,  Mathematical  Statistics  and  Theory 
of  Random  Functions  Dover,  N.Y.,  translation  of  1965  edition,  pp.  74,  116. 


891 


Tolstov,  G.P.,  1962  Fourier  Series,  Dover,  N.Y.,  pp.  173-177.  Reprint  of  1962  edition. 

Weiss,  G.H.,  1994  Aspects  and  Applications  of  the  Random  Walk,  North-Holland,  Amster¬ 
dam. 

Weiss,  S.H.  and  U.  Shmulei,  1987  Physica  146A  641-649.  Joint  Densities  for  Random 
Walks  in  the  Plane. 

Willie,  L.T.,  1987  Physica  141 A  509-523.  Joint  Distribution  Function  for  position  and 
Rotation  angle  in  Plane  Random  Walks. 

Wirsching,  RH.,  T.L.  Paez,  and  K.  Ortiz  1995  Random  Vibrations,  Theory  and  Practice, 
Wiley-Interscience,  N.Y.,  pp.  162-166. 


892 


A 

an 

Bn 

C{f) 

cih.h) 

E[N,] 

E[No^] 

iFi 

pEq 

f 

i 

3 

Jo 

k 

K 

Ly 

Ey 

m 

N 

Nf 

n 

Bviy) 

Py(^) 

pxYi^,y) 

S 

t 

T 

Y 

Y 
X 

aij 

r 

lij 

(f>n 


U 

a>n 


NOMENCLATURE 

amplitude,  peak,  or  envelope 

amplitude  of  the  nth  sine  wave,  a^,  >  0 

fitted  parameter  in  equation  (42) 

characteristic  function  with  parameter  / 

joint  characteristic  function  with  parameters  fi  and  /2 

expected  number  of  positive  peaks  per  unit  time 

expected  number  of  zero  crossing  with  positive  slope  per  unit  time 

confluent  hypergeometric  function  (Gradshteyn,  Ryzhik,  Jeffrey,  1994,  art.  9.210) 

generalized  hypergeometric  function  (Gradshteyn,  Ryzhik,  Jeffrey,  1994,  art. 

9.210) 

parameter  in  Fourier  transform 
integer  index 
imaginary  constant, 

Bessel  function  of  first  kind  and  zero  order 
integer  index 

complete  elliptic  integral  of  first  kind,  equation  (33a) 

+  ^2  +  ••  +  sum  of  amplitudes 
ujiai  +  uj2a2  +  -  •  +  sum  of  velocity  amplitudes 

integer  index 
number  of  terms  in  series 
cycles  to  failure 
integer  index,  n=l,2,..N 

cumulative  probability,  the  integral  of  Py{x)  from  x=— co  to  y 
probability  density  of  random  parameter  Y  evaluated  at  T  =  a: 
joint  probability  density  of  X  and  Y  evaluated  at  Y  =  y  and  X  =  x 
stress 

time,  0  <  t  <  T 

length  of  time  interval 

sum  of  N  modes  or  terms,  —Ly<Y  <  Ly 

first  derivative  with  respect  to  time  of  Y,  —Ly  <Y<  Ly 

a  random  variable 

dimensionless  coefficient,  equation  (33) 

gamma  function,  r[(2n  +  l)/2]  =  7r^/^2“’^(2n  -  1)!! 

dimensionless  coefficient,  equation  (36) 

Dirac  delta  function 
3.1415926.. 

XiX2.-xi\i,  product  of  terms 

phase  angle  of  the  nth  sine  wave,  a  uniformly  distributed  independent  random 
variable 

circular  frequency,  a  positive  (non  zero)  real  number 

circular  frequency  of  the  nth  term,  a  non  zero  integer  multiple  of  27r/T 


893 


Figure  1  Spectrum  of  vibration  of  a  component  on  a  turbojet  engine  cowling.  Note  the 
finite  number  of  distinct  peaks. 


894 


Figure  2  Sample  of  the  time  history  associated  with  the  spectrum  of  Figure  1.  Note  the 
signal  is  bounded,  irregular  and  quasi  sinusoidal. 


895 


-4.0  0  0.0  2.0 

MO,  OF  S.D. 


Figure  3  Probability  density  of  the  time  history  of  Figures  1  and  2.  Note  that  the  maximum 
values  do  not  exceed  plus  or  minus  2.5  standard  deviations. 


Yrms  *  (Probability  Density  of  Y) 


-3-2-10  1  2  3 

Y  /  Yrms 


- Normal  Distribution 

- Sine  Wave  Distribution 

o  Equation  .  N=10 
•  Equation  ,  N=1 


Figure  5  Normal  probability  density  (equation  18)  and  sine  wave  probability  density  (equa¬ 
tion  6)  in  comparison  with  results  of  numerical  integration  of  equation  (13)  for  N=1  and 
N=10. 


898 


Yrms*  (Probability  Density  of  Y) 


Yrms  (Probability  Density  of  A) 


- Two  Sine  Waves 

- Three  Sine  Waves 

. Four  Sine  Waves 


Figure  7  Probability  density  of  peaks  in  narrow  band  series  with  equal  amplitudes  (ai  = 
02..  =  1)  and  frequencies. 


900 


10* 


in'  {o'  10*  10*  10* 

FRTIGUE  LIFE.  CYCLES 

FIGURE  3.2.3.1.8(h).  Besi-fit  SI  N  curves  for  noiched.  K,  ~  4.0  of  2024-T3  aluminum  alloy  sheet, 
longitudinal  direction. 


Figure  8  Fatigue  curves  for  notched  2024-T3  aluminum  alloy  with  Kt=4.  MI1-HDBK-5G 
(1994,  p.  3-115) 


902 


ACOUSTIC  FATIGUE  I 


Strain  Power  Spectra  of  a  Thermally  Buckled  Plate 
in  Random  Vibration 

Jon  Lee  and  Ken  R.  Wentz 
Wright  Laboratory  (FIB) 

Wright-Patterson  AFB,  OH  45433,  USA 

Abstract 

Several  years  ago,  Ng  and  Wentz  reported  strain  power  spectra  measured 
at  the  mid-point  of  a  buckled  aluminum  plate  which  is  randomly  excited  by  an 
electrodynamic  shaker  attached  to  the  clamped-plate  boundary  fixture.  We 
attempt  to  explain  the  peculiar  features  in  strain  power  spectra  by  generating 
the  corresponding  power  spectra  by  the  numerical  simulation  of  a  single-mode 
equation  of  motion.  This  is  possible  because  the  essential  dynamics  takes  place 
in  the  frequency  range  just  around  and  below  the  primary  resonance  frequency. 

1.  Introduction 

For  high  performance  military  aircraft  and  future  high-speed  civil  transport 
planes,  certain  structural  skin  components  are  subjected  to  very  large  acoustic 
loads  in  an  elevated  thermal  environment  [1].  This  is  because  high-speed 
flights  call  for  a  very  powerful  propulsion  system  and  thereby  engendering 
acoustic  loads  in  the  anticipated  range  of  135-175  dB.  More  importantly, 
because  of  the  aerodynamic  heating  in  hypersonic  flights  and  the  modern  trend 
in  integrating  propulsion  sub-systems  into  the  overall  vehicular  configuration, 
some  structural  components  must  operate  at  high  temperatures  reaching  up  to 
1300°F.  Hence,  the  dual  effect  of  thermal  and  acoustic  loading  has  given  rise 
to  the  so-called  thermal-acoustic  structural  fatigue  [2,3]. 

Generally,  raising  the  plate  temperature  uniformly  but  with  an  immovable 
edge  boundary  constraint  would  result  in  thermal  buckling,  just  as  one  observes 
flexural  buckling  as  the  inplane  stress  along  plate  edges  is  increased  beyond  a 
certain  critical  value.  This  equivalence  has  been  recognized  [4,5]  and 
exploited  in  previous  analytical  and  experimental  investigations  of  the  thermal- 
acoustic  structural  fatigue  [6,7,8].  An  experimental  facility  for  thermal- 
acoustic  fatigue,  termed  the  Thermal  Acoustic  Fatigue  Apparatus,  was 
constructed  at  the  NASA  Langley  Research  Center  in  the  late  80’ s.  Under  the 
acoustic  loading  of  140-160  dB,  Ng  and  Clevenson  [9]  obtained  some  strain 
measurements  of  root-mean-square  value  and  power  spectral  density  (PSD)  on 
an  aluminum  plate  heated  up  to  250^.  Later,  Ng  and  Wentz  [10]  have 
repeated  the  heated  Aluminum  plate  experiment  but  by  randomly  exciting  the 
clamped-plate  boundary  fixture  by  a  shaker,  and  thereby  recovering  similar 
strain  measurements. 

It  should  be  noted  that  Ng  and  his  colleagues  [7,9,10]  were  the  first  to 
achieve  sufficient  plate  heating  to  induce  thermal  buckling  and  thus  observe  the 
erratic  snap-through  under  the  acoustic  or  shaker  excitations.  Here,  by  erratic 


903 


we  mean  that  a  snap-through  from  one  static  buckled  position  to  another  takes 
place  in  an  unpredictable  fashion.  We  reserve  the  adjective  chaotic  for  a  snap- 
through  occurring  under  the  deterministic  single-frequency  forcing  [1 1,12].  It 
has  already  been  observed  that  certain  of  the  buckled  plate  experiment  can  be 
explained,  at  least  qualitatively,  by  a  single-mode  model  of  plate  equations. 
This  is  also  validated  by  a  theoretical  analysis.  Indeed,  we  showed  that  a  single¬ 
mode  Fokker-Planck  formulation  can  predict  the  high-temperature  moment 
behavior  and  displacement  and  strain  histograms  of  thermally  buckled  plates, 
metallic  and  composite  [13,14]. 

In  retrospect,  a  single-mode  model  has  proven  more  useful  than  originally 
intended.  That  is,  the  single-mode  Fokker-Planck  formulation  of  an  isotropic 
plate  lends  itself  to  predicting  certain  statistics  of  composite  plates  which  are 
simulated  by  multimode  equations  or  tested  experimentally  by  multimode 
excitations.  For  a  refined  and  more  quantitative  comparison,  one  must  inject 
more  realism  into  dynamical  models  by  including  the  multimode  interactions. 
However,  before  giving  up  the  single-mode  plate  equation,  there  is  an 
important  problem  that  this  simple  model  is  well  suited  for  investigation.  That 
is,  prediction  of  the  strain  PSD  measurement  by  Ng  and  Wentz  [10].  As  we 
shall  see  in  Sec.  4,  the  strain  PSD  of  a  thermally  buckled  plate  exhibits  a  strong 
spectral  energy  transfer  toward  zero  frequency,  and  thereby  saturating 
frequency  range  well  below  the  primary  resonance  frequency.  This  downward 
spectral  energy  transfer  can  be  modeled  quite  adequately  by  the  single-mode 
plate  equation  without  necessitating  multimode  interactions. 

2.  Equation  of  motion  for  the  aluminum  plate  experiment 

By  the  Galerkin  procedure,  the  von  Karman-Chu-Herrmann  type  of  large- 
deflection  plate  equations  give  rise  to  infinitely  coupled  modal  equations  [15]. 
However,  much  has  been  learned  from  a  prototype  single-mode  equation  for 
displacement^  [13,14]. 

q  +  Pq  +  k„{l-s)q  +  aq^  =  g„  +  g{t),  (1) 

where  the  overhead  dot  denotes  d/dt  and  the  viscous  damping  coefficient  is 
P  =  2^^  with  damping  ratio  ^ .  For  the  clamped  plate,  we  have 

;i„=f(r‘'+2rV3  +  l), 

s = rji + (1  -M)  (1 + (r^+  ir^)  /6] , 
a  =  ^{(7^+r'^+2^i)  +  |(i-/i^)[T(r^+r'^)  +^(.r+r''T^ 

+  (r+47'‘)‘^  +  (47+  7"‘r^]} , 

&=  (r‘'+2r^/3  +i)Sjj6. 

Note  that  the  expressions  for  s  and  g„  are  specific  to  the  typical  temperature 


904 


variation  and  gradient  profiles  assumed  in  Ref.  [15].  Here,  7  =  b/a  is  the 
aspect  ratio  of  plate  sides  a  and  b,  and  fi  is  Poisson's  ratio.  The  uniform  plate 
temperature  is  measured  in  units  of  the  critical  buckling  temperature.  The 
maximum  temperature  variation  on  the  mid-plate  plane  is  denoted  by  and 
TJ5g  is  the  maximum  magnitude  of  temperature  gradient  across  the  plate 
thickness,  where  5^  and  5 ^  are  scale  factors.  Hence,  0  signifies  no 
temperature  variation  over  the  mid-plate  plane,  and  0  zero  temperature 
gradient  across  the  plate  thickness.  Finally,  g{t)  denotes  the  external  forcing. 

The  parameter  s  represents  thermal  expansion  due  to  both  the  uniform 
plate  temperature  rise  above  room  temperature  and  temperature  variation  over 
the  mid-plate  uniform  temperature.  The  combined  stiffness  k^(l  -  s)q  consists 
of  the  structural  stiffness  k^q  and  thermal  stiffness  -sk^q ,  which  cancel  each 
other  due  to  the  sign  difference.  It  is  positive  for  5  <1,  then  Eq.  (1)  has  the 
form  of  Duffing  oscillator  with  a  cubic  term  multiplied  by  a ,  which  represents 
geometric  nonlinearity  of  membrane  stretching.  For  s  >1  Eq.  (1)  reduces  to 
the  so-called  buckled-beam  equation  of  Holmes  [11]  with  a  negative  combined 
stiffness.  In  contrast,  denotes  thermal  moment  induced  by  a  temperature 
gradient  across  the  plate  thickness;  hence,  it  appears  in  the  right-hand  side  of 
Eq.  (1)  as  an  additional  forcing.  The  interplay  of  the  terms  involving  5,  a,  and 
g^  can  best  be  illustrated  by  the  potential  energy  [15] 

U{q)  =  -go q  +  k^(X-s) (fn  -H  a .  (2) 

Fig.  1  shows  that  V{c^  is  symmetric  when  g„  =  0.  For  s<l  it  has  a  single  well 
which  splits  into  a  double  well  as  s  exceeds  unity.  Note  that  the  distance 
between  the  twin  wells  increases  as  for  large  s  (Fig.  1(b)).  This 
interpretation  is  valid  approximately  for  go>^-  That  is,  a  positive  g^  lowers 
the  positive  side  potential  (^>0)  and  raises  the  negative  side  potential  {q<0), 
and  thereby  rendering  the  potential  energy  asymmetric. 

U(q)  ^(^1) 


Fig.  1  Potential  energy,  (a)  s<V,  (b)  ^  >1,  w  =  ^k^(s  -  l)/a  ,  d  =  -  l))V4a. 

( -  ^.=0;  ---  5„>0) 


905 


It  must  be  pointed  out  that  Eq.  (1)  is  dimensionless  and  involves  explicitly 
only  7  and  /i.  For  the  aluminum  plate  experiment  [10],  7=10  in./8  in.  and 
so  that  k^=Q3.9l  and  a =85. 33.  If  we  further  assume  5^-  0  for 

simplicity,  the  thermal  parameter  reduces  to  s=T^.  Previously,  Eq.  (1)  was 
used  for  the  investigation  of  stationary  Fokker-Planck  distribution  which 
involves  only  the  ratio  p!F,  where  F  is  the  constant  power  input  [13,14]. 
Hence,  nondimensionalization  has  indeed  spared  us  from  specifying  in  detail 
other  plate  parameters.  Things  are  however  different  in  numerical  simulation 
because  we  must  know  the  characteristic  scales  to  correctly  interpret  time- 
dependent  solutions.  By  retracing  the  derivation,  we  find  that  the  dimensionless 
quantities  in  Eq.  (1)  are  (Eq.  (IV.  1)  in  Ref.  [1]) 

q-qlh,  t-t/t*,  g  =  glg*,  (3) 

where  the  overhead  bar  denotes  the  physical  quantity.  Here,  the  plate  thickness 
h,  t*={b/Kf.y[ph/D,  and  g^=p(h/t*)^  are  the  characteristic  length,  time,  and 
force,  respectively  (p  =  mass  density,  D=Eh^f\2{\~p}) ,  £=  Young’s 
modulus  of  elasticity),  as  listed  in  Table  L  We  now  rewrite  Eq.  (1)  with  the 
numerical  coefficients  (Table  I).  _ 

q  +  0.0978?  +  23.91(1  -s)q  +  85.33?^  =  (4) 


where  g{t)  has  the  unit  of  psi. 

Table  I.  Parameter  values  for  the  aluminum  plate  experiment 

7,  « 

10in./8in.,  VoX  0.01,  23.910,  85.332 

5..  5, 

p 

o 

h,  f*,  g'^- 

0.05  in.,  3.305  lO'^sec.^,  5.806  1  O'’ psi^ 

(+)  p  =  0.098  Ib/in^  and  E  =1.03  10’  psi. 


3.  Monte-Carlo  simulation 

Because  of  5  =0,  Eq.  (4)  has  the  standard  form  of  Duffing  (s<l)  and 
Holmes  (s>l)  oscillators.  In  stead  of  a  single  frequency  for  forcing  g{t)  [11, 
12],  in  Monte-Carlo  simulation  all  forcing  frequencies  are  introduced  up  to  a 
preassigned  maximum  so  that  forcing  represents  a  plausible  physical 
realization.  Of  course,  particular  interest  here  is  a  constant  PSD.  We  shall 
begin  with  generation  of  a  time-series  for  random  processes  with  such  a  PSD. 

3.1  Random  forcing  time-series 

We  adopt  here  the  procedure  for  generating  a  time-series  of  Shinozuka 
and  Jan  [16],  which  has  been  used  for  a  oscillator  study  [17]  and  extensively 
for  structural  simulation  applications  by  Vaicaitis  [18,19].  Since  it  relies 
heavily  on  the  discrete  fast  Fourier  transforms,  such  as  FFTCF  and  FFTCB 
subroutines  of  the  IMSL  library,  it  is  more  expedient  to  describe  the  procedure 
operationally  rather  than  by  presenting  somewhat  terse  formulas.  Let  us 


906 


introduce  Nj:  frequency  coordinates  which  are  equally  spaced  in 
the  band  width  A/=/^3x/A^^.  Now,  the  task  is  to  generate  a  time-series  of  total 
time  T  that  can  resolve  up  to  .  Assume  T  is  also  divided  into  time 
coordinates  with  the  equal  time  interval  At=T/Nj,  From  the  time-frequency 
relation  r=l/A/,  we  find  ^At.  If  we  choose 

N 

Nf  =  (5) 

is  the  Nyquist  frequency,  consistent  with  our  original  definition  of  the 
upper  frequency  limit  of  resolution. 

A  random  time-series  with  a  constant  PSD  can  be  generated  in  the 
following  roundabout  way.  We  begin  by  assuming  that  we  already  have  a 
forcing  power  spectrum  ^g{f)  of  constant  magnitude  over  [0,/njax]-  Such  a 

PSD  may  be  represented  by  a  complex  array  A„= VC  exp(~27rz0„)  (n  =  1, 
Nf),  where  takes  a  random  value  distributed  uniformly  in  [0,  1].  Clearly  the 
magnitudes  of  are  C,  hence  We  then  enlarge  the  complex  array 

A„  by  padding  with  zeros  for  n  =Ny+l,  and  Fourier  transform  it  to 

obtain  a  complex  array  B^{n=  N^).  The  random  time-series  for  is 

now  given  by  the  real  part  of 

=  Real  part  of  (n  =  1, N^)  (6) 


As  it  turns  out,  when  g  „  is  padded  with  zeros  for  the  imaginary  components 
and  Fourier  transformed,  we  recover  the  original  array  A„  (w=l,  Nf)  with 


Since  the  spectrum  area  is  nothing  but  total  forcing  power 

<g^>  (say,  in  psi^),  we  can  relate  C  with  the  variance  <g^>  of  pressure 
fluctuations,  which  is  often  expressed  by  the  sound  pressure  level  (SPL)  in  dB, 
according  to  SPL=10  log<g2  >/p2,  where  p=2,9  10*^  psi.  Hence, 

c  =  - .  (7) 

/max 

Here,  Eqs.  (6)  and  (7)  defined  heuristically  are  meant  to  explain  the 
corresponding  formulas  (2)  and  (12)  in  Ref  [18]. 

For  the  numerical  simulation  we  first  note  that  the  resonance  frequency  of 
Eq.  (1)  is  /^=-y^/27r«  0.778  for  s  ~  0.  This  gives  the  dimensional  resonance 

frequency  fjt*~235.5  Hz  which  is  somewhat  larger  than  the  experimental 
217.7  Hz  (Fig.  3(a)).  As  shown  in  Table  II,  we  assign  (~9/r)  because 

the  electrodynamic  shaker  used  in  the  experiment  [10]  has  the  upper  frequency 
limit  2000  Hz. 


Table  II.  Dimensionless  parameter  values  for  the  numerical  simulation 

at,  N, _ 7,  8192,  4096 

At,  T  0.071.  585 


907 


3.2  Displacement  power  spectrum 

Under  a  random  time  integration  of  Eq.  (4)  yields  a  time-series  for  q^. 
We  first  comment  on  the  time  integration.  Although  there  are  special  solvers 
[17,20]  proposed  for  stochastic  ordinary  differential  equations  (ODEs),  we 
shall  use  here  the  Adams-Bashforth-Moulton  scheme  of  Shampine  and  Gordon 
[21],  which  has  been  implemented  in  DEABM  subroutine  of  the  SLATEK 
library.  Although  DEABM  has  been  developed  for  nonstochastic  ODEs,  its 
use  for  the  present  stochastic  problem  may  be  justified  in  part  by  that  one 
recovers  linearized  frequency  response  functions  by  the  numerical  simulation 
(Sec.  3.4).  Obviously,  this  does  not  say  anything  about  the  strongly  nonlinear 
problem  in  hand,  and  it  should  be  addressed  as  a  separate  issue.  In  any  event, 
DEABM  requires  the  absolute  and  relative  error  tolerances,  both  of  which  are 
set  at  no  larger  than  10“^  under  the  single-precision  algorithm  for  time 
integration.  Note  that  actual  integration  time  steps  are  chosen  by  the 
subroutine  itself,  commensurate  with  the  error  tolerances  requested.  Recall 
that  is  updated  at  every  time  interval  Ar,  and  we  linearly  interpolate  the 
forcing  value  within  A? . 

We  begin  time  integration  of  Eq.  (4)  from  the  initial  configuration  at  the 
bottom  of  the  single-well  potential,  ^(0)=p(0)=0,  for  5'  <1  and  the  positive 
side  double-well  potential,  q{0)=^kj,s~\)/a  andp(0)=0,  for  j  >1.  And  we 
continue  the  integration  up  to  T.  By  Fourier  transforming  time-series  q^,  we 
obtain  displacement  power  spectrum  0^(/).  This  process  of  integrating  and 
transforming  is  repeated  over  three  contiguous  time  ranges  of  7,  and  the 
successive  PSDs  are  compared  for  stationarity.  Since  it  is  roughly  stationary 
after  three  repetitions,  we  report  here  only  the  PSD  of  the  third  repetition. 
From  the  stationary  input-output  relation  [22]  where 

is  the  magnitude  of  system  frequency  response  function,  we  write 

=  (S) 

Since  O  (/)=C,  the  and  would  have  a  similar  functional 

s 

dependence  upon  /,  Hence,  we  call  them  both  the  displacement  PSD. 

3.3  Strain  power  spectrum 

Although  displacement  is  the  direct  output  of  numerical  simulation,  one 
measures  strain  rather  than  the  displacement  in  plate  experiment.  At  the 
present  level  of  plate  equation  formulation,  the  strain  e  is  given 

by  the  quadratic  relation 

e  =  +  C^q  +  C2(f‘  ■>  (9) 

where  C,-  are  given  at  the  middle  {x/a  =  y/b  =1/2)  of  a  clamped  plate  as  follows 
(Appendix  D  of  Ref  [13]) 


908 


c  = 


(/+2yV3+l)?;g, 

3(1+At)(y"+1) 


{l-li}- 


q  = 


8y^ 

3  ’ 


r  32  fy^  5n  (l-/iyV4) 

^  9  [2  16  2(y+y-‘f  (y+4y-‘)^  (4y  +  y‘fJ 


For  we  have  C^=0,  Q  =4.17,  and  Q  =2.77  (Table  I).  Hence,  Eq.  (9) 
engenders  only  the  linear  and  quadratic  transformations,  but  no  translation.  In 
any  event,  translation  has  no  effect  on  the  spectral  energy  contents.  By  Fourier 
transforming  time-series  (n=l,  A^^),  we  obtain  strain  power  spectrum 

.  Although  the  forcing  PSD  is  not  constant,  one  computes  the  forcing 
spectrum  ratio  as  in  Eq.  (8)  and  call  it  the  magnitude  square  of  strain  frequency 
response  function  for  the  lack  of  a  better  terminology. 

3.4  The  linear  oscillators 

For  the  pre-buckled  (5  <1)  linear  oscillator  (a=  0)  we  rewrite  Eq.  (4)  in 
standard  form  _ 

q  +  +  0)1(1  -s)q  =  (10) 

where  col=k^,  and  obtain 

I  H^(f)^  =  [(0)1(1  -s)-  +  (An^co^ffT'-  (11) 

As  shown  in  Fig.  2(a),  the  numerical  simulation  of  Eq.  (10)  recovers 
as  given  by  Eq.  (11)  over  the  entire  frequency  range.  Although  the  simulation 
of  Fig.  2(a)  was  carried  out  with  SPL=130  dB,  it  does  not  depend  on  SPL 
since  Eq.  (10)  is  linear.  Physically  speaking,  Eq.  (10)  oscillates  in  a  single-well 
potential  (Fig.  1(a)).  Since  the  potential  energy  has  two  wells  (Fig.  1(b))  for 
s  >1,  we  linearize  Eq.  (1)  around  the  positive  side  potential  well  by  the 
transformation  q=q'+^k^{s-l)la .  Hence,  the  corresponding  linear  oscillator 

is 


(a)  (b) 


Fig.  2  Linear  frequency  response  functions,  (a)  Displacement;  (b)  Strain 
(j  =  0;  - Numerical  simulation;  •  Eq.  (U)) 


909 


q'  +  2^0)^  q’  +  Icolis  -V)q'  =  (12) 

s 

In  parallel  to  Eq.  (1 1),  the  frequency  response  function  of  a  post-buckled  plate 

I  =  [(^-colis  -  1)  -  4;rV^)2  +  .  (13) 

The  resonance  frequency  f=co^^2{s-l)/27J:  of  a  post-buckled  (s  >1)  plate 
should  be  compared  with  f=co^^2{\~s)/2n  of  the  pre-buckled  (j  <1)  plate. 

Now,  for  the  linear  oscillators  we  see  that  is  also  given  by  Eq.  (1 1) 

and  (13)  for  ^  <1  and  >1,  respectively  (Fig.  2(b)).  This  is  because  the  spectral 
energy  distribution  is  not  at  ail  affected  by  a  linear  transformation. 

4.  Displacement  and  strain  power  spectra 

As  we  shall  see  in  Sec  4.1,  the  experimental  strain  PSD  exhibits  downward 
spectral  energy  transfer  toward  zero  frequency,  so  that  there  is  a  considerable 
spectral  energy  buildup  below  the  resonance  frequence  as  SPL  is  raised. 
Moreover,  it  also  involves  an  upward  spectral  energy  transfer  which  then 
contributes  to  both  the  increased  resonance  frequency  and  broadened 
resonance  frequency  peak.  Since  spectral  energy  transfers  take  place  around 
and  below  the  primary  resonance  frequency,  it  is  possible  to  depict  the 
downward  and  upward  spectral  energy  transfers  by  the  numerical  simulation  of 
Eq.  (4)  without  necessitating  multimode  interactions.  We  shall  first  discuss  the 
characteristic  features  of  experimental  strain  PSDs. 

4. 1  Experimental  strain  PSD 

Of  the  spectra  reported  in  Ref.  [10],  we  consider  the  following  two  sets. 
One  is  the  nonthermal  set  (^=0)  consisting  of  two  PSDs  of  small  and  large 
SPLs.  The  other  is  the  post-buckled  set  (5=1.7)  of  four  PSDs.  For  the 
convenience  of  readers,  we  have  reproduced  in  Figs.  3  and  4  the  selected  PSDs 
from  Ref.  [10]  by  limiting  the  upper  frequency  to  600  Hz,  and  the  pertinent 
data  are  summarized  in  Table  m. 


Table  in.  Strain  power  spectra  of  experiment  and  numerical  simulation 


Fig.  4(a)  Fig.  6a 


Fig.  4(b)  Fig.  6b 


Fig.  4(c)  Fig.  6c 
Fig.  4(d)  Fig.  6d (*) 


Fig.  7 
Fig.  8 
Fig.  9 


Fig.  10 


(*)  Computed  from  the  acceleration  a  measured  in  units  of  g. 


910 


Fig.  4  Experimental  strain  PSD  =1.7).  (a)  130.1dB;  (b)  142dB;  (c)  151.5dB;  (d)  154.6dB 

The  following  observations  are  drawn  from  the  experimental  PSDs.  First, 
for  the  nonthermal  plate 

Figure  3 fa):  Compare  the  measured  strain  fr-2\l  Hz  with  the  theoretical 
displacement 235  Hz  of  Eq.  (4).  Note  that  a  small  spectral  energy  peak  is 
found  at  467  Hz  which  is  about  twice  (-2.15)  the  strain  value. 


911 


Figure  Sfb):  With  SPL~150  dB  the  strain  increases  to  240  Hz  and  the 
spectral  width  at  the  half  resonance  peak  has  nearly  doubled.  The  spectral 
energy  buildups  at  zero  and  515  Hz  are  more  noticeable  than  in  Fig.  3(a). 
Again,  515  Hz  is  about  twice  (-2.15)  the  primary  strain  f,. 


(a)  (b) 


f  f 


Fig.  5  Numerical  simulation  results  under  .y=0  and  SPL=130  dB. 

(a)  Displacement  ( - simulation,*  Eq.  (11));  (b)  Strain  ( - simulation,  •  Eq.  (11)); 

(c)  PSD  averaged  over  12  frequency  intervals  ( - displacement,  — •  —  strain); 

(d)  Strain  PSD. 


Next,  for  the  thermally  buckled  plate 

Figure  4(a):  The  primary  strain  fr=227  Hz  should  be  compared  with  the 
theoretical  displacement  /^=279  Hz  of  Eq.  (13).  A  second  spectral  energy 
peak  is  found  at  537  Hz,  much  larger  than  twice  (-2.37)  the  primary  strain  /^. 
Figure  4rb):  Here,  the  spectral  energy  buildup  is  most  significant  at  zero 
frequency.  Besides,  there  appear  two  spectral  energy  humps  at  100  and  183 
Hz,  below  the  primary  strain  =  227  Hz  of  Fig.  4(a).  Discounting  the  zero- 
frequency  spectral  peak,  PSD  may  be  approximated  by  a  straight  line  in  the 
semi-log  plot,  hence  it  is  of  an  exponential  form  up  to  400  Hz. 


Figure  4rc):  The  zero-frequency  peak  is  followed  by  a  single  spectral  energy 
hump  at  115  Hz.  Again,  PSD  can  be  approximated  by  a  straight  line  and  its 
slope  is  roughly  the  same  as  in  Fig.  4(b). 

Figure  4rd):  A  major  spectral  energy  peak  emerges  at  130  Hz,  followed  by  a 
minor  one  at  350  Hz.  Theoverall  spectral  energy  level  is  raised  so  that  the 
magnitude  of  PSD  ranges  over  only  two  decades  in  the  figure. 

In  Figs.  4(b)-(d)  we  have  ignored  the  spectral  energy  peaks  at  around  500 
Hz,  for  they  are  not  related  to  the  first  plate  mode  under  consideration.  This  is 
further  supported  by  the  simulation  evidence  to  be  discussed  presently. 


4.2  Numerical  simulation  results 

After  choosing  .y  =  0  or  1.7,  we 
are  left  with  SPL  yet  to  be  specified.  o 
Ideally,  one  would  like  to  carry  out  the 
numerical  simulation  of  Eq.  (4)  by 
using  SPL  of  the  plate  experiment  'm-z 
(Table  III)  and  thus  generate  strain  ^ 

PSDs  which  are  in  agreement  with 
Figs.  3  and  4.  Not  surprisingly,  the  _4 
reality  is  less  than  ideal.  An  obvious 
reason  that  this  cannot  be  done  is  that 
the  forcing  energy  input  is  fed  into  all  f 

plate  modes  being  excited  in  Hg.  6  PSD  averaged  over  12  frequency 
experiment,  whereas  the  forcing  (j=0,  SPL=138dB) 

energy  excites  only  one  mode  in  the  - displacement;  -•-strain 

numerical  simulation.  Consequently,  SPL  for  the  numerical  simulation  should 
be  less  than  the  experimental  SPL,  but  we  do  not  know  a  priori  how  much 
less.  We  therefore  choose  a  SPL  to  bring  about  qualitative  agreements 
between  the  single-mode  simulation  and  multimode  experiment.  As  anticipated, 
the  simulation  SPLs  (Table  HI)  are  consistently  smaller  than  the  experimental 
values. 

The  numerical  simulation  results  are  shown  in  Figs.  5-6  for  5  =  0  and  Figs. 
7-10  for  s  =1.7.  Actually  each  figure  has  four  frames,  denoted  by  (a)-(d). 
First,  frames  (a)  and  (b)  depict  and  Since  they  are  very 

jagged  at  large  SPLs,  we  average  the  spectral  energy  over  12  frequency 
intervals  and  present  both  of  the  smoothed-out  frequency  response  functions  in 
the  same  frame  (c).  Lastly,  frame  (d)  shows  Og(/)  itself  Since  there  is  no 
qualitative  difference  between  <E>g(/)  and  we  shall  call  them  both  the 

strain  PSD.  We  present  all  four  frames  (a)— (d)  of  Figs.  5  and  7,  but  only  the 
frame  (c)  of  Figs.  6,  8,  9  and  10  here  for  the  lack  of  space. 

First,  for  the  nonthermal  plate 

Figure  5:  The  simulated  is  closely  approximated  by  Eq.  (11)  with  f  = 

236  Hz.  Note  that  is  also  approximated  by  Eq.  (11)  for  all  frequencies 


913 


Fig.  7  Numerical  simulation  results  under  j=1.7  and  SPL=129  dB. 

(a)  Displacement  ( - simulation,*  Eq.  (13));  (b)  Strain  ( - simulation,  •  Eq.  (13)); 

(c)  PSD  averaged  over  12  frequency  intervals  ( - displacement,  — •  strain); 

(d)  Strain  PSD. 


but  zero  and  476  Hz,  where  the  strain  spectral  energy  piles  up  due  to  the 
quadratic  transformation  (9).  Since  476  Hz  is  nearly  twice  (-2.02)  the  primary 
/^,  strain  spectral  energy  buildups  are  due  to  the  sum  and  difference  of  two 

nearly  equal  frequencies,  ±  /2,  where/i==/2^/^. 

Figure  6:  The  primary  strain  is  shifted  slightly  upward  to  253  Hz  and  the 

spectral  width  at  half  resonance  peak  is  50%  wider  than  that  of  Fig.  5(c).  The 
spectral  energy  builds  up  at  525  Hz  which  is  roughly  twice  (-2.08)  the  /^.  At 
SPL=138  dB  we  find  that  the  strain  spectral  energy  hump  at  525  Hz  is  about  2 
decades  below  the  resonance  frequency  peak,  as  was  in  Fig.  3(b). 

Now,  for  the  thermally  buckled  plate 

Figure  7:  The  simulated  and  are  weU  approximated  by  Eq. 

(13)  around  /^=270  Hz  which  is  a  litde  below  the  linearized  /^=279  Hz. 
Unlike  in  Fig.  5  for  5=0,  both  and  l/7^(/)F  show  spectral  energy 

building  up  significantly  near  zero  and  543  Hz  which  is  twice  (-2.01)  the  /^. 


914 


2p — ! - 1 - ! - 1 - 1 - T— n - 1 - r 


^ _ 1 - 1 - 1 - < - 1 - > - 1 - ^ - 1 

0  300  600 


f 

Fig.  8  PSD  averaged  over  12  frequency 
intervals  (j  =1.7,  SPL=138  dB) 

- displacement;  — •  —  strain 


Note  that  in  Fig.  7(a)  the  spectral 
energy  hump  at  543  Hz  is  about  3 
decades  below  the  primary  frequency 
peak,  as  was  in  Fig.  4(a). 

Figure  8:  After  a  large  zero-frequency 
peak,  two  spectral  energy  humps 
appear  at  131  Hz  and  236  Hz.  Note 
that  the  ratios  of  these  frequencies  to 
the  /,  (131/279  -0.47  and  236/279  - 
0.85)  are  comparable  with  the  same 
ratios  (100/227  -0.44  and  183/227  - 
0.81)  found  in  Fig.  4(b).  Excluding 
the  zero-frequency  peak,  the  overall 
strain  PSD  is  a  straight  line,  hence  of 
an  exponential  form,  as  in  Fig.  4(b), 


Figure  9:  The  zero-frequency  spectral  peak  is  followed  by  a  single  major 
energy  hump  at  154  Hz.  The  ratio  of  this  to  the  (154/279  -0.56)  is 
somewhat  larger  than  the  ratio  (115/227  -0.51)  in  Fig.  4(c).  The  strain  PSD 
can  also  be  approximated  by  a  straight  line  over  the  entire  frequency  range,  and 
Figs.  8  and  9  seem  to  have  the  same  slope  when  fitted  by  straight  lines. 

Figure  10:  The  spectral  magnitude  of  is  larger  than  that  of  in 

the  frequency  range  above  300  Hz.  The  choice  of  SPL=146  dB  was  based  on 
that  the  PSD  magnitude  around  300  Hz  is  about  2  decades  below  the  main 
spectral  peak  magnitude  at  180  Hz,  thus  emulating  Fig.  4(d). 

All  in  all,  by  numerical  simulations  we  have  successfully  reproduced  the 
peculiar  features  in  the  two  sets  of  strain  PSDs  observed  experimentally  under  5 
=  0  and  1.7. 


Fig.  9  PSD  averaged  over  12  frequency 
intervals  (s  =1.7,  SPL=143  dB) 

- displacement;  — • — strain 


Fig.  10  PSD  averaged  over  12  frequency 
intervals  (j  =1.7,  SPL=146  dB) 

- displacement;  — • — strain 


915 


5.  Concluding  remarks 

At  low  SPL  the  nonthermal  {s=  0)  and  post-buckled  (^=1.7)  plates  appear 
to  have  a  similar  PSD.  However,  this  appearance  is  quite  deceptive  in  that  the 
nonthermal  plate  motion  is  in  a  single-well  potential,  so  that  PSD  does  not 
change  qualitatively  as  SPL  is  raised.  On  the  other  hand,  the  trajectory  of  a 
post-buckled  plate  is  in  one  of  the  two  potential  energy  wells  when  SPL  is  very 
small.  However,  as  we  raise  SPL  such  a  plate  motion  can  no  longer  be 
contained  in  a  potential  well,  and  hence  it  encircles  either  one  or  both  of  the 
potential  wells  in  an  erratic  manner.  This  is  why  the  experimentally  observed 
and  numerically  simulated  strain  PSDs  of  a  post-buckled  plate  exhibit 
qualitative  changes  with  the  increasing  SPL,  and  thereby  reflect  the  erratic 
snap-through  plate  motion.  A  quantitative  analysis  of  snap-through  dynamics 
will  be  presented  elsewhere. 

Lastly,  we  wish  to  point  out  that  a  PSD  of  straigh-line  form  in  the  semi-log 
plot  was  observed  in  a  Holmes  oscillator  when  trajectories  are  superposed 
randomly  near  the  figure-eight  separatrix  [23]. 

Acknowledgments 

Correspondence  and  conversations  with  Chung  Fi  Ng,  Chuh  Mei,  Rimas 
Vaicaitis,  and  Jay  Robinson  are  sincerely  appreciated.  We  also  wish  to  thank 
the  referees  for  their  helpful  suggestions  to  improve  the  readability  of  this 
paper. 


References 

1.  Lee,  J.,  Large-Amplitude  Plate  Vibration  in  an  Elevated  Thermal 
Environment,  WL-TR-92-3049,  Wright  Lab.,  Wright-Patterson  AFB,  OH, 
June,  1992. 

2.  Jacobson,  M.J.  and  Maurer,  O.F.,  Oil  canning  of  metallic  panels  in 
thermal-acoustic  environment,  AIAA  Paper  74-982,  Aug.,  1974. 

3.  Jacobson,  M.J.,  Sonic  fatigue  of  advanced  composite  panels  in  thermal 
environments,  J.  Aircraft,  1983,  20,  282-288. 

4.  Bisplinghoff,  R.L.  and  Pian,  T.H.H.,  On  the  vibrations  of  thermally 
buckled  bars  and  plates,  in  Proc.  9th  Inter.  Congr.  of  Appl.  Mech., 
Brussels,  1957,  7,  307-318. 

5.  Tseng,  W.-Y.,  Nonlinear  vibration  of  straight  and  buckled  beams  under 
harmonic  excitation,  AFOSR  69-2157TR,  Air  Force  Office  of  Scientific 
Research,  Arlington,  VA,  Nov.,  1969. 

6.  Seide,  P.  and  Adami,  C.,  Dynamic  stability  of  beams  in  a  combined 
thermal-acoustic  environment,  AFWAL-TR-83-3072,  Flight  Dynamics 
Lab.,  Wright-Patterson  AFB,  OH,  Oct.,  1983. 

7.  Ng,  C.F.,  Nonlinear  and  snap-through  responses  of  curved  panels  to 
intense  acoustic  excitation,  /.  Aircraft,  1989,  26,  281-288. 

8.  Robinson,  J.H.  and  Brown,  S.A.,  Chaotic  structural  acoustic  response  of  a 
milled  aluminum  panel,  36th  Structures,  Structural  Dynamics,  and 


916 


Material  Conference,  AIAA-95-1301-CP,  New  Orleans,  LA,  1240-1250, 
Apr.  10-13,  1995. 

9.  Ng,  C.F.  and  Clevenson,  S.  A.,  High-intensity  acoustic  tests  of  a  thermally 
stressed  plate,  J,  Aircraft,  1991,  28,  275-281.. 

10.  Ng,  C.F.  and  Wentz,  K.R.,  The  prediction  and  measurement  of  thermo¬ 
acoustic  response  of  plate  structures,  31st  Structures,  Structural 
Dynamics,  and  Material  Conference,  AIAA-90-0988-CP,  Long  Beach, 
CA,  1832-1838,  Apr.  2-4, 1990. 

11.  Holmes,  P.,  A  nonlinear  oscillator  with  a  strange  attractor,  Phil.  Trans., 
Roy.  Soc.  of  London,  1979,  292A,  419-448. 

12.  Dowell,  E.H.  and  Pezeski,  C.,  On  the  understanding  of  chaos  in  Duffings 
equation  including  a  comparison  with  experiment,  J.  Appl.  Mech.,  1986, 
53,  5-9. 

13.  Lee,  J.,  Random  vibration  of  thermally  buckled  plates:  I  Zero  temperature 
gradient  across  the  plate  thickness,  in  Progress  in  Aeronautics  and 
Astronautics,  168,  Aerospace  Thermal  Structures  and  Materials  for  a 
New  Era,  Ed.  E.A.  Thornton,  AIAA,  Washington,  DC,  1995.  41-67. 

14.  Lee,  J.,  Random  vibration  of  thermally  buckled  plates:  n  Nonzero 
temperature  gradient  across  the  plate  thickness,  to  appear  in  J.  Vib.  and 
Control,  1997. 

15.  Lee,  J.,  Large-amplitude  plate  vibration  in  an  elevated  thermal 
environment,  Mech.  Rev.,  1993,  46,  S242-254. 

16.  Shinozuka,  M.  and  Jan,  C.-M.,  Digital  simulation  of  random  processes 
and  its  applications,  J.  Sound  and  Vib. 1912,  25,  1 1 1-128. 

17.  Chiu,  H.M.  and  Hsu,  C.S.,  A  cell  mapping  method  for  nonlinear 
deterministic  and  stochastic  systems  -  Part  II:  Examples  of  application,  J. 
Appl.  Mech.,  1986,  53,  702-710. 

18.  Vaicaitis,  R.,  Nonlinear  response  and  sonic  fatigue  of  national  aerospace 
space  plane  surface  panels,  J.  Aircraft,  1994,  31,  10-18. 

19.  Vaicaitis,  R.,  Response  of  Composite  Panels  Under  Severe  Thermo- 
Acoustic  Loads,  Report  TR-94-05,  Aerospace  Structures  Information  and 
Analysis  Center,  Wright-Patterson  AFB,  OH,  Feb.,  1994. 

20.  Kasdin,  N.J.,  Runge-Kutta  algorithms  for  the  numerical  integration  of 
stochastic  differential  equations,  J.  Guidance,  Control,  and  Dynamics, 
1995,18,  114-120. 

21.  Shampine,  L.F.  and  Gordon,  M.K.,  Computer  solution  of  ordinary 
differential  equations,  1975,  Freeman,  San  Francisco. 

22.  Lin,  Y.K.,  Probabilistic  theory  of  structural  dynamics,  Robert  E.  Krieger 
Publishing,  1976,  Huntington,  NY. 

23.  Brunsden,  V.,  Cortell,  J.  and  Holmes,  P.J.,  Power  spectra  of  chaotic 
vibrations  of  a  buckled  beam,  J.  Sound  and  Vib.,  1989, 130,  1-25. 


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ENHANCED  CAPABELITIES  OF  THE  NASA  LANGLEY 
THERMAL  ACOUSTIC  FATIGUE  APPARATUS 

Stephen  A.  Rizzi  and  Travis  L.  Turner 
Structural  Acoustics  Branch 
NASA  Langley  Research  Center 
Hampton,  VA  23681-0001 

ABSTRACT 

This  paper  presents  newly  enhanced  acoustic  capabilities  of  the  Thermal 
Acoustic  Fatigue  Apparatus  at  the  NASA  Langley  Research  Center.  The 
facility  is  a  progressive  wave  tube  used  for  sonic  fatigue  testing  of  aerospace 
structures.  Acoustic  measurements  for  each  of  the  six  facility  configurations 
are  shown  and  comparisons  with  projected  performance  are  made. 

INTRODUCTION 

The  design  of  supersonic  and  hypersonic  vehicle  stmctures  presents  a 
significant  challenge  to  the  airframe  analyst  because  of  the  wide  variety  and 
severity  of  environmental  conditions.  One  of  the  more  demanding  of  these  is 
the  high  intensity  noise  produced  by  the  propulsion  system  and  turbulent 
boundary  layer  [1].  Complicating  effects  include  aero-thermal  loads  due  to 
boundary  layer  and  local  shock  interactions,  static  mechanical  preloads,  and 
panel  flutter.  Because  of  the  difficulty  in  accurately  predicting  the  dynamic 
response  and  fatigue  of  structures  subject  to  these  conditions,  experimental 
testing  is  often  the  only  means  of  design  validation.  One  of  the  more  common 
means  of  simulating  the  thermal- vibro-acoustic  environment  is  through  the  use 
of  a  progressive  wave  tube.  The  progressive  wave  tube  facility  at  NASA 
Langley  Research  Center,  known  as  the  Thermal  Acoustic  Fatigue  Apparatus 
(TAFA),  has  been  used  in  the  past  to  support  development  of  the  thermal 
protection  system  for  the  Space  Shuttle  and  National  Aerospace  Plane  [2].  It 
is  presently  being  used  for  sonic  fatigue  studies  of  the  wing  strake 
subcomponents  on  the  High  Speed  Civil  Transport  [3]. 

The  capabilities  of  the  TAFA  were  previously  documented  by  Clevenson  and 
Daniels  [4].  The  system  was  driven  by  two  Wyle  WAS  3000  airstream 
modulators  which  provided  an  overall  sound  pressure  level  range  of  between 
125  and  165  dB  and  a  useful  frequency  range  of  50-200  Hz.  A  360  kW  quartz 
lamp  bank  provided  radiant  heat  with  a  peak  heat  flux  of  54  W/cm^.  A 
schematic  of  the  facility  is  shown  in  Figure  1.  Representative  spectra  and 
coherence  plots  are  shown  in  Figures  2  and  3.  Since  that  time,  the  facility  has 
undergone  significant  enhancements  designed  to  improve  its  acoustic 
capabilities;  the  heating  capabilities  were  not  changed.  The  objectives  of  the 
enhancements  were  to  increase  the  maximum  overall  sound  pressure  level 
(OASPL)  to  178  dB,  increase  the  frequency  bandwidth  to  500  Hz  and  improve 
the  uniformity  of  the  sound  pressure  field  in  the  test  section.  This  paper 


919 


documents  the  new  capabilities  of  the  TAFA  and  makes  comparisons  with  the 
projected  performance. 


Figure  1:  Schematic  of  the  old  TAFA  facility. 


Figure  2:  Test  section  spectra  of  the  Figure  3:  Test  section  coherence  of  the 
old  TAFA  facility.  old  TAFA  facility. 


FACILITY  DESCRIPTION 

In  order  to  meet  the  design  objectives,  extensive  modifications  were  made  to 
the  sound  generation  system  and  to  the  wave  tube  itself.  A  theoretical  increase 
of  6  dB  OASPL  was  projected  by  designing  the  system  to  utilize  eight  WAS 
3000  air  modulators  compared  to  the  two  used  in  the  previous  system.  A 
further  increase  of  nearly  5  dB  was  expected  by  designing  the  test  section  to 
accommodate  removable  water-cooled  insert  channels  which  reduced  its  cross- 
sectional  area  from  1.9m  x  0.33m  to  0.66m  x  0.33m.  The  frequency  range  was 
increased  through  the  use  of  a  longer  horn  design  with  a  lower  (15  Hz  vs.  27 
Hz  in  the  old  facility)  cut-off  frequency,  use  of  insert  channels  in  the  test 
section  to  shift  the  frequency  of  significant  standing  waves  above  500  Hz,  and 
design  of  facility  sidewall  stmctures  with  resonances  above  1000  Hz.  The 
uniformity  of  the  sound  pressure  field  in  the  test  section  was  improved  through 
several  means.  A  new,  smooth  exponential  horn  was  designed  to  avoid  the 
impedance  mismatches  of  the  old  design.  To  minimize  the  effect  of 
uncorrelated,  broadband  noise  (which  develops  as  a  byproduct  of  the  sound 


920 


generation  system),  a  unique  design  was  adopted  which  allows  for  the  use  of 
either  two-,  four-,  or  eight-modulators.  When  testing  at  the  lower  excitation 
levels  for  example,  a  two-modulator  configuration  might  be  used  to  achieve  a 
lower  background  level  over  that  of  the  four-  or  eight-modulator 
configurations.  In  doing  so,  the  dynamic  range  is  extended.  Lastly,  a  catenoidal 
design  for  the  termination  section  was  used  to  smoothly  expand  from  the  test 
section. 

Schematics  of  the  facility  in  the  three  full  test  section  configurations  are  shown 
in  Figures  4-6.  In  the  two-modulator  configuration,  the  2  x  4  transition  cart 
acts  to  block  all  but  two  of  the  eight  modulators.  The  facility  is  converted 
from  the  two-  to  four-modulator  configuration  by  the  removal  of  the  2  x  4 
transition  cart  and  connection  of  two  additional  modulators.  In  doing  so,  the 
modulator  transition  cart  slides  forward  and  thereby  maintains  the  continuous 
exponential  expansion  of  the  duct.  In  the  four-modulator  configuration,  the  4 
X  8  transition  cart  acts  to  block  the  two  upper  and  two  lower  modulators. 
Removal  of  this  component  and  connection  of  the  four  additional  modulators 
converts  the  facility  to  the  eight-modulator  configuration.  Again,  the 
continuous  exponential  expansion  is  maintained  as  the  modulator  transition 
cart  slides  forward. 


Figure  4:  Two-modulator  full  test  section  configuration. 


921 


Figure  5:  Four-modulator  full  test  section  configuration. 


Figure  6:  Eight-modulator  full  test  section  configuration. 


Schematics  of  the  three  reduced  test  section  configurations  are  shown  in 
Figures  7-9.  In  these  configurations,  the  horn  cart  is  discarded  and  the  horn 
transition  cart  mates  directly  to  the  test  section.  Water-cooled  inserts  are  used 
in  the  test  section  to  reduce  its  cross-sectional  area.  Upper  and  lower  inserts  in 
the  termination  section  are  used  to  smoothly  transition  the  duct  area  to  the  full 
dimension  at  the  exit.  Conversion  from  the  two-  to  the  four-modular 
configuration  and  from  the  four-  to  the  eight-modulator  configuration  is  again 
accomplished  through  removal  of  the  2  x  4  and  4x8  transition  carts, 
respectively. 


922 


HORN  TTUNSTTION  SECTION 


Figure  7:  Two-modulator  reduced  test  section  configuration. 


Figure  8:  Four-modulator  reduced  test  section  configuration. 

TEST  PROCEDURE 


Measurements  were  taken  for  several  conditions  in  each  of  the  six  facility 
configurations.  Each  modulator  was  supplied  with  air  at  a  pressure  of  207  kPa 
(mass  flow  rate  of  approximately  8.4  kg/s)  and  was  electrically  driven  with  the 
same  broadband  (40-500  Hz)  signal.  Acoustic  pressures  were  measured  at 
several  locations  along  the  length  of  the  progressive  wave  tube  using  B&K 
model  4136  microphones  and  Kulite  model  MIC-190-HT  pressure  transducers, 
see  Table  1.  The  positive  x-direction  is  defined  in  the  two-modulator  full 
configuration  (from  the  modulator  exit)  along  the  direction  of  the  duct.  The 
positive  y-direction  is  taken  vertically  from  the  horizontal  centerline  of  the 


923 


HORN  TRIWSmOH  SECTION 


MODULATOR  TRANSITION 
FLEMSie  HOSE 


adapter  puts  assembly 

TEST  SECTION 

ADAPTER  plate  assembly 
TERM1ASAT10N  SECTION 


horn  TRANSITION  CART 


Figure  9:  Eight-modulator  reduced  test  section  configuration. 

duct  and  the  positive  z-direction  is  defined  from  the  left  sidewall  of  the  duct  as 
one  looks  downstream. 


Table  1:  Kulite  (K)  and  microphone  (M)  locations  of  acoustic  measurements. 


Loc. 

Description 

Type 

Coordinate  (m) 

1 

Test  Sect.  Horizontal  Centerline  Upstream 

K 

7.75,  0,  0 

2 

Test  Sect.  Horizontal  Centerline  Downstream 

K 

8.71,  0,  0 

5 

Test  Sect.  Vertical  Centerline  Top 

M 

8,23,  0.3,  0 

15 

Test  Sect.  HorizontaWertical  Centerline 

K 

8,23, 0,  0 

25 

Test  Sect.  Vertical  Centerline  Bottom 

M 

8.23,  -0.3,  0 

28 

2x4  HorizontaWertical  Centerline 

M 

2.19,  0,0 

29 

4x8  Horizontal  Centerline,  %  Downstream 

M 

3.66, 0,  0 

30 

Horn  Tran,  Hor.  Centerline,  %  Downstream 

M 

4.75, 0,  0 

35 

Termination  HorizontaWertical  Centerline 

M 

12.46, 0,  0.17 

The  acoustic  pressure  at  location  1  was  used  as  a  reference  measurement  for 
shaping  the  input  spectrum  and  for  establishing  the  nominal  overall  sound 
pressure  level  for  each  test  condition.  For  each  configuration,  the  input 
spectrum  to  the  air  modulators  was  manually  shaped  through  frequency 
equalization  to  produce  a  nearly  flat  spectrum  at  the  reference  pressure 
transducer.  Data  was  acquired  at  the  noise  floor  level  (flow  noise  only)  and  at 
overall  levels  above  the  noise  floor  in  6  dB  increments  (as  measured  at  the 
reference  location)  up  to  the  maximum  achievable.  Thirty-two  seconds  of 
time  data  were  collected  at  a  sampling  rate  of  4096  samples/s  for  each 
transducer  in  each  test  condition.  Post-processing  of  the  time  data  was 
performed  to  generate  averaged  spectra  and  coherence  functions  with  a  1-Hz 
frequency  resolution. 


924 


RESULTS 

For  each  facility  configuration,  plots  of  the  following  quantities  are  presented: 
normalized  input  spectrum  to  the  air  modulators,  minimum  to  maximum 
sound  pressure  levels  at  the  reference  location,  maximum  sound  pressure 
levels  in  the  test  section,  maximum  sound  pressure  levels  upstream  and 
downstream  of  the  test  section,  and  vertical  and  horizontal  coherence  in  the 
test  section.  The  minimum  levels  in  each  case  correspond  to  the  background 
noise  produced  by  the  airflow  through  the  modulators. 

Normalized  input  voltage  spectra  to  each  modulator  for  each  configuration  are 
shown  in  Figures  10,  15,  20,  25  and  30.  These  spectra  were  generated  to 
achieve  as  flat  an  output  spectrum  as  possible  at  the  reference  location  for  the 
frequency  range  of  interest  (40-200  Hz  for  the  full  section,  40-500  Hz  for  the 
reduced  section).  As  expected,  the  significant  difference  between  the  full  and 
reduced  configurations  is  seen  in  the  high  (>200  Hz)  frequency  content. 

Figure  11  shows  a  background  noise  level  of  126  dB  (the  lowest  of  all 
configurations)  for  the  two-modulator  full  test  section  configuration.  Nearly 
flat  spectra  are  observed  below  210  Hz  for  levels  above  130  dB,  giving  a 
dynamic  range  of  about  32  dB.  The  flat  spectrum  shape  is  a  significant 
improvement  over  the  performance  of  the  old  configuration  as  shown  in  Figure 
2.  Standing  waves  are  evident  at  frequencies  of  210,  340  and  480  Hz.  For  this 
reason,  the  full  section  operation  is  limited  to  less  than  210  Hz  or  to  the  220- 
330  and  370-480  Hz  frequency  bands.  The  effect  of  standing  waves  are 
explored  in  further  depth  in  the  next  section.  The  spectra  in  Figure  12  indicate 
a  nearly  uniform  distribution  in  the  x-direction  throughout  the  test  section.  It  is 
interesting  to  note  that  Figure  13  shows  no  sign  of  standing  waves  upstream  of 
the  test  section,  confirming  that  the  cause  is  associated  with  the  test  section. 
Lastly,  a  near  perfect  coherence  between  upstream  and  downstream,  and  upper 
and  lower  test  section  locations  is  shown  in  Figure  14  for  frequencies  between 
40  and  210  Hz.  Again,  this  is  a  significant  improvement  over  the  performance 
of  the  old  configuration  (Figure  3). 


Figure  10:  Normalized  input  spectrum  Figure  11:  Min  to  max  SPL  at  location 
(2-modulator  full).  1  (2-modulator  full). 


925 


Figure  12:  SPL  in  test  section  at  max  Figure  15:  Normalized  input  spectrum 
level  (2-modulator  full).  (4-modulator  full). 


Figure  13:  SPL  along  length  of  TAFA  Figure  16:  Min  to  max  SPL  at  location 
(2-modulator  full).  1  (4-modulator  full). 


Figure  14:  Test  section  coherence  (2-  Figure  17:  SPL  in  test  section  at  max 
modulator  full).  level  (4-modulator  full). 

The  four-modulator  full  configuration  exhibits  similar  behavior  as  the  two- 
modulator  full  configuration  as  seen  in  Figures  16-19.  The  lowest  level  at 
which  a  uniform  spectrum  is  achieved  is  137  dB,  giving  a  dynamic  range  of 
roughly  30  dB  in  this  configuration.  Lastly,  the  eight-modulator  full 


926 


configuration  results,  shown  in  Figures  21-24,  indicate  a  noise  floor  of  about 
142  dB  and  dynamic  range  of  22  dB. 


Frequency,  Hz  Frequency,  Hz 


Figure  18:  SPL  along  length  of  TAFA  Figure  21:  Min  to  max  SPL  at  location 
(4-modulator  full).  1  (8-modulator  full). 


Frequency,  Hz  Frequency,  Hz 

Figure  19:  Test  section  coherence  (4-  Figure  22:  SPL  in  test  section  at  max 
modulator  full).  level  (8-moduiator  full). 


Frequency,  Hz  Frequency,  Hz 


Figure  20:  Normalized  input  spectrum  Figure  23:  SPL  along  length  of  TAFA 
(8-modulator  full).  (8-modulator  full). 


927 


Frequency,  Hz  Frequency,  Hz 


Figure  24:  Test  section  coherence  (8-  Figure  27:  SPL  in  test  section  at  max 
modulator  full).  level  (2-modulator  reduced). 


Figure  25:  Normalized  input  spectrum  Figure  28:  SPL  along  length  of  TAFA 
(2-modulator  reduced).  (2-modulator  reduced). 


Figure  26:  Min  to  max  SPL  at  location  Figure  29:  Test  section  coherence  (2- 
1  (2-modulator  reduced).  modulator  reduced). 

The  reduced  test  section  configurations  are  used  to  increase  the  frequency 
range  and  maximum  sound  pressure  level  in  the  test  section.  Results  for  the 
two-modulator  reduced  configuration,  shown  in  Figures  26-29,  indicate  a 
nearly  flat  spectrum  between  40  and  480  Hz,  a  noise  floor  of  129  dB  and  a 
dynamic  range  of  about  28  dB.  Coherence  in  the  test  section  is  nearly  unity 


928 


over  this  frequency  range.  This  represents  a  significant  improvement  over  the 
old  facility  configuration.  Results  of  similar  quality  indicate  a  d5mamic  range 
of  roughly  26  and  29  dB  for  the  four-  (Figures  31-34)  and  eight-modulator 
(Figures  36-39)  configurations,  respectively.  Note  that  the  coherence  for  these 
configurations  is  slightly  reduced  at  the  high  frequencies,  but  is  still  very  good 
out  to  480  Hz. 


Figure  30:  Normalized  input  spectrum  Figure  33:  SPL  along  length  of  TAFA 
(4-modulator  reduced).  (4-modulator  reduced). 


160  r 


80 


OASPLs:  134.1,  135.5,  142.1,  147.6, 
153.9,  160.1,  165.9, 167.9 


1.0 

!  1 

1  1 

Loc  1,  Loc  2 
Loc  5,  Loc  25 


400 


100 


200  300 

Frequency,  Hz 


400 


500 


Figure  3 1 :  Min  to  max  SPL  at  location  Figure  34:  Test  section  coherence  (4- 
1  (4-modulator  reduced).  modulator  reduced). 


Figure  32:  SPL  in  test  section  at  max  Figure  35:  Normalized  input  spectrum 
level  (4-modulator  reduced).  (8-modulator  reduced). 


929 


80 


OASPLs;  134.1, 135.7,  141.9, 148.3, 
154.1.  160.0,  165.9. 170.5 


80 


Loc30 

Loci 


100  200  300  400  500  100  200  300  400  500 

Frequency.  Hz  Frequency,  Hz 

Figure  36:  Min  to  max  SPL  at  location  Figure  38:  SPL  along  length  of  TAFA 
i  (8-modulator  reduced).  (8-modulator  reduced). 


Figure  37:  SPL  in  test  section  at  max  Figure  39:  Test  section  coherence  (8- 
level  (8-modulator  reduced).  modulator  reduced). 


Table  2  presents  a  summary  of  the  maximum  average  OASPL  for  each  facility 
configuration.  In  each  case,  the  number  of  active  modulators  were  run  at 
maximum  power  as  an  independent  group  (independently  for  the  single 
modulator  case)  and  the  results  averaged.  For  example,  results  for  one  active 
modulator  were  obtained  by  running  each  modulator  individually  and 
averaging  the  resulting  pressures. 


Table  2:  Summary  of  maximum  average  overall  sound  pressure  levels  (dB). 


Number  of  Active  Modulators 

1 

2 

4 

8 

2-Modulator  Red. 

1 

1  j 

2-Modulator  Full 

i 

4-Modulator  Red. 

159.1 

— 

MSM 

4-Modulator  Full 

155.6 

161.2 

8-Modulator  Red. 

158.4 

— 

mmm 

171.7* 

8-Modulator  Full 

153.0 

158.4 

164.5 

170.0 

^Pressure  scaled  by  ^7?  from  7-modu 

ator  run 

DISCUSSION 


In  this  section,  limiting  behaviors  of  the  full  and  reduced  test  section 
configurations  are  explored  and  the  effect  of  test  section  inserts,  modulator 
coupling  and  wave  tube  performance  are  discussed. 

Limiting  Behaviors 

The  auto-spectra  from  the  full  test  section  configurations  exhibit  sharp 
reductions  in  level  at  approximately  210,  340,  and  480  Hz.  This  behavior 
corresponds  to  measurements  near  nodes  of  vertical  (height)  standing  waves  in 
the  test  section  portion  of  the  wave  tube.  Table  3  summarizes  theoretical, 
resonant  frequencies  and  corresponding  modal  indices  of  the  test  section  duct 
resonances  within  the  excitation  bandwidth.  The  modal  indices  m  and  n 
correspond  to  half  wavelengths  in  the  vertical  and  transverse  (width)  directions 
of  the  cross  section,  respectively.  There  are  several  resonances  that  may  be 
excited  below  500  Hz,  but  only  three  of  these  appear  to  be  significant  at  the 
test  section  transducer  locations  (about  the  horizontal  centerline).  Because  of 
the  presence  of  air  flow  in  the  facility  and  lack  of  measurements  in  the  cross 
section,  it  is  difficult  to  correlate  the  experimental  and  theoretical  modes. 
Measurements  of  the  acoustic  pressure  at  several  locations  in  a  cross-section  of 
the  duct  will  be  necessary  to  fully  characterize  the  resonant  behavior.  It  is 
sufficient  to  say  that  the  usable  frequency  range  in  the  full  test  section 
configurations  is  approximately  40-210  Hz  near  the  horizontal  centerline. 
Acoustic  pressure  auto-spectra  from  the  reduced  test  section  configurations  are 
essentially  flat  to  almost  500  Hz.  This  is  due  to  the  fact  that  only  two 
resonances  are  within  the  excitation  bandwidth  for  this  configuration,  see 
Table  3.  A  sharp  reduction  is  noted  in  the  vicinity  of  480  Hz.  Although  the 
(m=l,  n=0)  resonance  does  not  appear  to  be  significant,  close  inspection  of  the 
data  (not  shown)  indicates  its  presence.  Therefore,  the  usable  frequency  range 
for  the  reduced  test  section  configurations  is  approximately  40-500  Hz. 

Table  3:  Theoretical  resonant  frequencies  of  test  section  duct  modes  in  Hz. 


Performance  of  Test  Section  Configurations 

For  constant  input  acoustic  power,  the  change  from  full  to  reduced  test  section 
configurations  should  theoretically  result  in  a  4,7  dB  increase  in  OASPL. 
However,  Table  2  shows  that  increases  of  only  2.1  (e.g.  164.3-162.2),  0.9,  and 
1.7  dB  were  realized  for  the  two-,  four-  and  eight-modulator  configurations. 
The  system  efficiency  (actual/expected  mean-square  pressure)  of  the  two-, 
four-  and  eight-modulator  reduced  configurations  is  38,  40  and  44  percent, 
respectively,  compared  with  51,  63  and  62  percent  for  the  two-,  four-  and 
eight-modulator  full  configurations.  The  expected  pressure  is  calculated  based 
upon  a  input-scaled  value  of  the  rated  acoustic  power  of  the  WAS  3000 
modulator  assuming  incoherent  sources  (3  dB  per  doubling).  In  general,  the 
full  section  efficiency  is  greater  than  the  corresponding  reduced  section 
efficiency.  While  the  reason  for  this  phenomena  is  not  known,  it  is 
conjectured  that  the  lack  of  expansion  in  the  reduced  configurations  limits  the 
development  of  plane  waves.  Therefore,  phase  and  amplitude  mismatches 
between  acoustic  sources  may  be  accentuated. 

Modulator  Coupling  Performance 

A  simplified  waveguide  analysis  for  coherent,  phase-matched  sources  predicts 
increases  in  OASPL  as  shown  in  Table  4.  Measured  performance  gains  were 
less  than  predicted  because  of  the  assumptions  of  the  waveguide  analysis 
(inactive  source  area  treated  as  hard  wall),  and  possible  reductions  due  to 
phase  differences  between  modulators  and  non-parallel  wave  fironts  at  the  exit 
of  the  modulator  cart,  see  Figures  4-9.  The  latter  effect  is  due  to  different 
angles  of  inclination  of  the  sources  relative  to  the  axis  of  the  wave  tube.  The 
greater  gains  achieved  in  the  full  test  section  configurations  support  the  above 
contention  that  they  are  more  efficient  than  the  reduced  configurations  in 
combining  the  acoustic  sources. 


Table  4:  Change  in  SPL  (dB)  from  1  to  max.  number  of  active  modulators. 


Configuration 

A  SPL  fi:om  1  Active  Mod.  (Meas/Pred) 

2-Modulator  Red.  (2  active  mods.) 

3.9/6.53 

2-Modulator  Full  (2  active  mods.) 

5.5/6.53 

4-Modulator  Red.  (4  active  mods.) 

8.8  / 13.98 

4-Modulator  Full  (4  active  mods.) 

11.4/13.98 

8-Moduiator  Red.  (8  active  mods.) 

13.3/22.10 

8-Moduiator  Full  (8  active  mods.) 

17.0/22.10 

Wave  Tube  Performance 


A  change  in  configuration  from  the  two-  to  the  four-modulator  configurations, 
and  from  the  four-  to  the  eight-modulator  configurations,  will  result  in  an 
incremental  increase  of  3  dB  in  OASPL  if  the  individual  sources  are  phase- 
matched.  This  is  due  to  a  pure  doubling  of  the  power  without  any  change  in 
the  radiation  impedance  of  the  individual  sources.  For  the  reduced 
configurations,  a  3.6  and  3.8  dB  increase  are  observed,  respectively.  A  4.8  and 


932 


3.0  dB  increase  are  observed  for  the  full  configurations,  respectively.  Note 
that  a  greater  than  3  dB  increase  is  possible  when  the  higher  modulator 
configuration  (for  example,  the  four-modulator  reduced  configuration)  is  less 
susceptible  than  the  lower  modulator  configuration  (the  two-modulator 
reduced  configuration)  to  phase  mismatches  between  modulators.  This  seems 
plausible  because  any  such  mismatches  are  averaged  over  a  larger  number  of 
sources. 

SUMMARY 

Modifications  to  the  NASA  Langley  TAFA  facility  resulted  in  significant 
improvements  in  the  quality  and  magnitude  of  the  acoustic  excitation  over  the 
previous  facility.  The  maximum  OASPL  was  increased  by  over  6  dB  (vs  the 
previous  165  dB)  with  a  nearly  flat  spectrum  between  40-210  and  40-480  Hz 
for  the  full  and  reduced  test  section  configurations,  respectively.  In  addition, 
the  coherence  over  the  test  section  was  excellent.  These  improvements, 
however,  did  not  meet  the  objective  for  a  maximum  OASPL  of  178  dB. 

There  are  several  reasons  why  the  maximum  OASPL  did  not  meet  the 
objectives,  including  a  lack  of  expansion  in  the  reduced  configurations  and 
phase  differences  between  modulators.  A  detailed  computational  analysis 
would  be  desirable  to  indicate  the  source  of  the  inefficiencies  and  to  help 
identify  possible  means  of  increasing  the  overall  system  performance. 
ACKNOWLEDGEMENTS 

The  authors  wish  to  thank  Mr.  H.  Stanley  Hogge  and  Mr.  George  A.  Parker  for 
their  support  in  configuring  and  running  the  facility.  We  wish  to  also  thank 
Mr.  James  D.  Johnston,  Jr.  of  NASA  Johnson  Space  Center  for  loan  of  four 
Wyle  air  modulators. 

REFERENCES 

1.  Maestrello,  L.,  Radiation  from  a  Panel  Response  to  a  Supersonic 
Turbulent  Boundary  Layer,  Journal  of  Sound  and  Vibration,  1969, 
10(2),  pp.  261-295. 

2.  Pozefsky,  P.,  Blevins,  R.D.,  and  Langanelli,  A.L.,  Thermal-Vibro- 
Acoustic  Loads  and  Fatigue  of  Hypersonic  Flight  Vehicle  Structure, 
AFWAL-TR-89-3014, 

3.  Williams,  L.J.,  HSCT  Research  Gathers  Speed,  Aerospace  America, 
April  1995,  pp.  32-37. 

4.  Clevenson,  S.A.  and  Daniels,  E.F.,  Capabilities  of  the  Thermal 
Acoustic  Fatigue  Apparatus,  NASA  TM  104106,  February  1992. 


933 


SONIC  FATIGUE  CHARACTERISTICS  OF  HIGH  TEMPERATURE  MATERIALS  AND 
STRUaURES  FOR  HYPERSONIC  FLIGHT  VEHICLE  APPLICATIONS- 


Dr.  I.  Holehouse,  Staff  Specialist, 
Rohr  Inc.,  Chula  Vista,  California 


1.  INTRODUCTION  SUWiARY 

A  combined  analytical  and  experimental  program  was  conducted  to  investigate 
thermal -acoustic  loads,  structural  response,  and  fatigue  characteristics  of 
skin  panels  for  a  generic  hypersonic  flight  vehicle.  Aerothermal  and 
aeroacoustic  loads  were  analytically  quantified  by  extrapolating  existing 
data  to  high  Mach  number  vehicle  ascent  trajectories.  Finite-element 
thermal  and  sonic  fatigue  analyses  were  performed  on  critically  affected 
skin  panels.  High  temperature  random  fatigue  shaker  tests  were  performed 
on  candidate  material  coupons  and  skin-stiffener  joint  subelements  to 
determine  their  random-fatigue  strength  at  high  temperatures.  These  were 
followed  by  high  temperature  sonic  fatigue  tests  of  stiffened-skin  panels 
in  a  progressive  wave  tube.  The  primary  materials  investigated  were 
carbon-carbon  and  silicon-carbide  refractory  composites,  titanium  metal 
matrix  composites  and  advanced  titanium  alloys.  This  paper  reports  on  the 
experimental  work  and  compares  measured  frequencies  and  acoustically 
induced  response  levels  with  analytically  predicted  values. 

The  coupon  shaker  test  data  were  used  to  generate  material  random  fatigue 
"S-N"  curves  at  temperatures  up  to  980°C.  The  joint  subelements  provided 
data  to  determine  the  effects  on  fatigue  life  of  skin-stiffener  joining 
methods.  The  PWT  sonic  fatigue  panel  tests  generated  response  and  fatigue 
life  data  on  representative  built-up  skin  panel  design  configurations  at 
temperatures  up  to  925“C  and  sound  pressure  levels  up  to  165  dB.  These^ 
data  are  used  in  determining  the  response  strains  and  frequencies  of  skin 
panel  designs  when  subjected  to  combined  thermal -acoustic  loading  and  to 
identify  modes  of  failure  and  weaknesses  in  design  details  that  affect _ 
sonic  fatigue  life.  Sonic  fatigue  analyses  of  selected  test  panel  design 
configurations  using  finite-element  techniques  were  also  performed  and 
related  to  the  experimental  results.  Acoustically  induced  random  stresses 
were  analytically  determined  on  a  mode-by-mode  basis  using  finite  element 
generated  mode  shapes  and  an  analytical  procedure  that  extends  Miles 
approach  to  include  multi-modal  effects  and  the  spatial  characteristics  of 
both  the  structural  modes  and  the  impinging  sound  field. 

The  paper  also  describes  the  instrumentation  development  work  performed  in 
order  to  obtain  reliable  strain  measurements  at  temperatures  in  excess  of 
conventional  strain  gauge  capabilities.  This  work  focused  primarily  on  the 
use  of  recently  developed  high  temperature  (350"C  to  1000“C)  strain  gauges, 
laser  Doppler  vibrometers,  high  temperature  capacitance  displacement 
probes,  and  the  determination  of  strain-displacement  relationships  to 
facilitate  the  use  of  double  integrated  accelerometer  data  to  derive  strain 
levels. 


935 


This  work  was  funded  by  the  USAF  Flight  Dynamics  Laboratory  (Kenneth  R. 
Wentz,  Project  Engineer).  The  complete  program  report  is  contained  in 
References  1  and  2. 


2.  HIGH  TEMPERATURE  STRAIN  MEASUREMENTS 

Conventional  adhesively  bonded  strain  gauge  installations  are  temperature 
limited  to  approximately  350°C.  In  order  to  achieve  strain  measurements  at 
higher  temperatures,  up  to  QSO^C,  ceramic  layers  and  coatings  were  used  to 
both  attach  strain  gauges  and  to  thermally  protect  them.  However,  such 
strain  gauge  installations  are  very  sensitive  to  process  parameters  which 
often  need  varying  depending  upon  the  test  specimen  material.  Coated 
carbon-carbon  is  a  particularly  difficult  material  to  adhere  to  due  to  its 
material  characteristics  and  relatively  rough  surface  texture.  Carbon- 
carbon  also  has  a  near  zero  coefficient  of  thermal  expansion  which  presents 
attachment  and  fixturing  problems  in  a  high  temperature  environment. 

When  high  test  temperatures  either  preclude  or  make  problematic  the  use  of 
strain  gauges,  an  alternative  technique  for  obtaining  strain  levels  is  to 
measure  displacements  and  then  determine  strain  levels  using  strain- 
displacement  ratios.  Strain  is  directly  proportional  to  displacement  for  a 
given  deflected  shape,  or  mode  shape,  regardless  of  changes  in  the  elastic 
modulus  of  the  specimen  material  as  it  is  heated.  Consequently,  if  the 
deflected  shape  does  not  change  significantly  with  temperature,  high 
temperature  test  strain  levels  can  be  determined  from  room  temperature 
strain  and  displacement  measurements  in  combination  with  displacement 
measurements  made  at  the  test  temperature. 

This  measurement  technique  facilitates  the  use  of  non- contacting 
transducers  which  can  be  located  away  from  the  heated  area,  such  as 
capacitance  displacement  probes  or  Laser  Doppler  Vibrometers  (LDV).  LDVs 
actually  measure  surface  velocity  but  their  signal  outputs  can  be  readily 
integrated  and  displayed  as  displacement.  Accelerometers  can  also  be  used 
to  measure  displacement  by  double  integrating  their  signal  output. 

However,  since  accelerometers  require  surface  contact  they  have  to  either 
withstand,  or  be  protected  from,  the  thermal  environment.  When  this  is  not 
readily  achievable,  it  is  sometimes  possible  to  install  an  accelerometer  at 
a  location  on  the  test  specimen  or  fixturing  where  the  temperature  is 
within  its  operating  range,  providing  the  displacement  response  at  the 
point  of  measurement  is  fully  coherent  with  the  strain  response  at  the 
required  location. 

The  displacement  range  limitations  of  the  LDV  and  capacitance  probes 
available  to  the  program  resulted  in  having  to  use  double-integrated 
accelerometer  outputs  to  measure  displacements  at  room  temperature  and  at 
the  test  temperature.  Conventional  strain  gauges  were  used  to  measure 
strains  at  room  temperature.  In  order  to  confirm  that  the  strain- 
displacement  ratios  were  unaffected  by  temperature,  limited  high 
temperature  strains  were  measured  at  temperatures  up  to  980* C.  Once  the 
strain-displacement  ratio  for  a  given  specimen  type  was  determined,  air¬ 
cooled  accelerometers  were  used  to  determine  high  temperature  test  strain 
levels.  The  level  of  measurement  accuracy  of  this  technique  was  estimated 
to  be  within  10  percent. 


936 


The  most  successful  strain  measurements  made  at  980“ C  utilized  a  ceramic 
flame  spray  installation  of  an  HFN  type  free  filament  gauge.  This  gauge 
installation  included  the  use  of  silicon-carbide  (SiC)  cement  as  a  base 
coat  for  the  gage,  applied  over  a  1-inch  square  area  of  a  lightly  sanded 
carbon-carbon  surface  substrate.  Lead  wire  attachments  to  the  gauge  were 
made  with  standard  Nichrome  ribbon  wire  anchored  to  the  specimen  with  SiC 
cement.  With  this  gauge  installation,  it  was  possible  to  make  dynamic 
strain  measurements  for  short  periods  of  time  at  980“C. 


3.  RANDOM  FATIGUE  SHAKER  TESTS 

The  instrumented  test  specimens  were  mounted  in  a  duckbill  fixture  and  the 
specimen/fixture  assembly  then  enclosed  in  a  furnace.  An  opening  in  the 
furnace  allows  the  specimen  tip  to  protrude  out  in  order  to  accorrmodate  the 
air-cooled  tip  accelerometer.  Figure  1  shows  strain  gauge  locations  and 
fixturing  for  material  coupon  and  joint  subelement  specimens. 

The  test  procedure  comprised  a  room  temperature  sine-sweep  in  order  to 
identify  the  fundamental  mode  and  its  natural  frequency,  one-third  octave 
random  loading  at  room  temperature  centered  around  the  fundamental  natural 
frequency  and  one-third  octave  random  endurance  testing  at  the  required 
test  temperature  and  load  level. 

Twelve  inhibited  carbon-carbon  material  coupons  generated  usable  S-N  data, 
eleven  at  980“C  and  one  at  650“C.  S-N  data  points  were  also  generated  at 
980°C  for  two  integral  joint  and  two  mechanically  fastened  joint 
subelements.  Fixturing  problems  and  specimen  availability  limited  the 
number  of  S-N  data  points  generated.  Figure  2  shows  the  random  fatigue  S-N 
data  points  with  joint  subelement  data  points  superimposed.  The  random 
fatigue  endurance  level  for  the  material  coupons,  extrapolated  from  10  to 
10®  cycles,  is  approximately  320  microstrain  rms.  The  integral  joint 
subelements  did  not  fail  at  the  strain  gauge  locations;  consequently,  the 
actual  maximum  strain  levels  were  higher  than  those  shown  on  Figure  2. 
Taking  this  into  account,  it  appears  that  the  integral  joints  have  a 
fatigue  endurance  level  of  greater  than  one-half  of  that  for  the  material 
coupons.  The  mechanically  fastened  joint  subelements  exhibited  fatigue 
strength  comparable  to  that  of  the  material  coupons.  These  results 
indicate  that  carbon-carbon  joints  and  attachments  methods  are  not 
critically  limiting  factors  in  the  structural  applications  of  inhibited 
carbon-carbon.  Figure  3  shows  a  representative  example  of  the  strain 
amplitude  and  peak  strain  amplitude  probability  density  functions  at  room 
temperature  for  a  material  coupon  specimen.  The  "peak"  function  can  be 
seen  to  approximate  a  Rayleigh  distribution,  as  it  should  for  a  Gaussian 
random  process. 

Random  fatigue  S-N  data  were  also  generated  for  enhanced  silicon-carbide 
composites  (SiC/SiC)  including  thermally  exposed  specimens  (160  hours  at 
980“ C),  titanium  metal  matrix  composites  (TMC)  utilizing  Ti  15-3  and  Beta 
21S  titanium  matrix  materials,  titanium  aluminide  (super  alpha  two), 
titanium  6-2-4-2,  titanium  6-2-4-2-$i  (including  thermally  exposed 
specimens)  and  Ti-1100.  The  fatigue  endurance  levels  are  shown  in  Table  1. 
Also  shown  in  Table  1  are  S-N  data  points  for  uninhibited  carbon-carbon 
generated  on  a  previous  program  (Reference  3). 


937 


TABLE  1.  SUMMARY  OF  RANDOM  FATIGUE  ENDURANCE  LEVELS. 


material 

TEMPERATURE 

ENDURANCE  LEVEL  CORRESPONDING  TO  10® 
CYCLES:  OVERALL  RMS  STRAIN  (MICROSTRAIN) 

material  COUPONS 

SUBELEMENTS 

INHIBITED  CARBON-CARBON 

1800“F  (980"C) 

320 

integral  JOINTS  >  160 

BOLTED  JOINTS  320 

5  PLY 

1800T  (980"C) 

100 

- 

UNINHlbl 1 tU 
CARBON-CARBON 

1ft  PI  Y 

ISOO'-F  igSOT) 

150 

_ 

*5  PLY 

1000‘'F  I540"C) 

100 

- 

18  PLY 

lOOO'F  (540-C) 

450 

- 

5  PLY 

ROOM  TEMPERATURE 

550 

- 

18  PLY 

ROOM  TEMPERATURE 

450 

- 

ENHANCED 

SiC/SiC 

NON-EXPOSED 

1800"F  r980“C) 

450 

- 

THERMALLY  EXPOSED 

iaOO“F  f980‘'C) 

300 

- 

1000“F  (540*0 

520 

DIFFUSION-BONDED  HAT- 
STIFFENED  =  520 

n  IML 

ROOM  TEMPERATURE 

2250 

- 

beta  21S  TMC 

200 

510 

- 

TITANIUM  ALUMINIDE 

/ciinCD  ftl  DUIA  TUn^ 

ROOM  TEMPERATURE 

410 

{  rvui  1  iri  •  * 

Ti-6242-Si 

735 

Hilllll 

LIO  BONDED  HONEYCOMB 
BEAM  =  388 

■■ 

WELDED  JOINT  =  400 

T-;  nnn 

- 

Ti  6-2-4-2 

ROOM  TEMPERATURE 

675 

- 

866MISC/039-T1.IH 

12-02-96 


938 


Figure  4  shows  random  fatigue  S-N  curves  for  the  materials  tested 
superimposed  on  one  graph  for  comparison  purposes.  The  Ti  15-3  TMC  data 
are  not  shown  since  this  was  a  concept  demonstrator  material  utilizing  a 
Ti  15-3  matrix  material  for  producibility  reasons.  Ti  15-3  does  not  have 
the  temperature  capability  for  hypersonic  vehicle  applications.  Titanium 
aluminide  data  are  not  shown  due  to  its  brittle  material  characteristics 
making  it  unsuitable  for  sonic  fatigue  design  critical  structures.  Ti-1100 
S-N  data  were  very  similar  to  the  non-exposed  Ti  6-2-4-2-Si  and  are  not 
shown.  Ti  6-2-4-2  coupons  were  only  tested  at  room  temperature  before 
beingVeplaced  by  Ti  6-2-4-2-Si,  which  has  higher  structural  temperature 
capabilities. 

The  fatigue  curves  in  Figure  4  show  inhibited  carbon-carbon  to  have  higher 
fatigue  strength  at  980°C  than  does  its  uninhibited  counterpart.  Inhibited 
carbon-carbon  also  has  greater  resistance  to  oxidation  at  high 
temperatures. 

Although  unexposed  enhanced  SiC/SiC  had  greater  random  fatigue  strength  at 
980“C  than  did  inhibited  carbon-carbon,  the  two  materials  exhibited  similar 
strength  at  temperature  after  allowing  for  thermal  exposure.  However, 
SiC/SiC  has  a  maximum  temperature  capability  of  1100  to  1200°C  compared  to 
1700  to  1900“C  for  carbon-carbon. 

The  Beta  21S  TMC  material  demonstrated  resonable  fatigue  strength  at  815°C 
and  the  Ti  6-2-4-2-Si  specimens  exhibited  high  fatigue  strength  at  620°C  to 
650“C. 


4.  SONIC  FATIGUE  PANEL  TESTS 

These  tests  were  performed  in  Rohr's  high  temperature  progressive-wave  tube 
(PWT)  test  facility.  The  facility  is  capable  of  generating  overal 1^ sound 
pressure  levels  of  165  to  168  dB  at  temperatures  up  to  925 “C  to  980“ C, 
depending  upon  the  test  panel  configuration  and  material. 

Three  rib-stiffened  carbon-carbon  panels  and  a  monolithic  hat-stiffened 
Beta  21S  TMC  panel  were  subjected  to  sonic  fatigue  testing.  Response 
strains  were  measured  on  the  four  panels  over  a  range  of  incrementally 
increasing  sound  pressure  levels  (140  to  165  dB)  at  room  temperature.  One 
carbon-carbon  panel  was  subjected  to  sonic  fatigue  testing  at  room 
temperature  and  the  other  two  tested  at  925“ C.  The  TMC  panel  was  endurance 
tested  at  815“C.  Figures  5  and  6  show  a  carbon-carbon  panel  and  its 
fixturing  installed  in  the  PWT.  The  panels  were  attached  to  the  fixture 
via  flexures  in  order  to  allow  for  differences  in  the  thermal  expansion  of 
the  panel  and  fixture  materials.  Structural  details  of  the  panels  and 
instrumentation  locations  are  given  in  References  1  and  2. 

The  three  carbon-carbon  panel  configurations  encompassed  two  skin 
thicknesses  and  two  stiffener  spacings  as  follows: 

Panel  1:  3  skin  bays,  6  in.  by  20  in.  by  0.11  in.  thick 

Panel  2:  2  skin  bays,  9  in.  by  20  in.  by  0.11  in.  thick 

Panel  3:  3  skin  bays,  6  in.  by  20  in.  by  0.17  in.  thick 


939 


Table  2  summarizes  the  measured  room  temperature  frequencies  and  strain 
response  levels: 


TABLE  2.  ROOM  TEMPERATURE  RESPONSE  OF  TEST  PANELS- 


TEST 

PANEL 

FREQUENCY  OF 
IN-PHASE 
MODE  (Hz) 

OASPL 

(dB) 

OVERALL  RMS  STRAINS 
rMICROSTRAIN) 

EDGE  OF 
SKIN  BAY 

CENTER  OF 
SKIN  BAY 

CARBON-CARBON  NO.  1 

267 

165 

305 

149 

155  &  171 

145 

126 

39 

191 

59 

165 

558* 

173* 

CARBON-CARBON  NO.  3 

423 

165 

69 

127 

BETA  21$  TMC 

241 

165 

HIGHEST  STRAIN  =  287 

AT  PANEL  CENTER  ON 

STIFFENER  CAP 

*  EXTRAPOLATED  ON  THE  BASIS  OF  TUE  STRAIN  RESPONSE  WITH  SPL  FOR 
PANELS  1  AND  3. 


Panel  1  was  subjected  to  165  dB  at  room  temperature  for  10  hours  at  which 
point  cracks  developed  at  the  ends  of  the  stiffeners.  The  frequency 
dropped  slightly  during  the  ten  hour  test  resulting  in  the  number  of  cycles 
to  failure  being  approximately  9  million. 

Panel  2  was  endurance  tested  at  925“C  at  150,  155  and  160  dB  for  3-1/2 
hours  at  each  level,  followed  by  one  hour  at  165  dB.  At  this  point,  cracks 
were  observed  at  the  ends  of  the  stiffeners,  similar  to  the  cracks  in 
Panel  1. 

Panel  3  was  endurance  tested  at  925“C  and  165  dB  for  10  hours  without  any 
damage  to  the  panel. 

The  TMC  panel  was  endurance  tested  at  815° C  and  165  dB  for  3-1/2  hours  at 
which  time  cracks  were  observed  in  two  stiffener  caps  at  the  panel  center. 

The  high  test  temperatures  for  Panels  2  and  3  and  the  TMC  panel  precluded 
attaching  an  accelerometer  directly  to  the  panel  surface,  even  with  air 
cooling.  This  prevented  the  direct  measurement  of  panel  displacements  at 
925° C.  In  order  to  attempt  to  estimate  the  high  temperature  endurance  test 
strain  levels,  a  temperature  survey  was  performed  on  the  panel  fixturing 
with  Panel  3  installed  in  order  to  determine  an  acceptable  location  for  an 
accelerometer.  An  accelerometer  at  the  selected  fixture  location  tracked 
linearly  with  the  highest  reading  strain  gauges  during  a  room  temperature 
response  survey.  The  coherence  between  the  fixture  accelerometer  and  the 
panel  strain  gauges  was  0.9  in  the  frequency  range  of  panel  response. 


940 


Having  established  a  coherent  strain  displacement  relationship  at  room 
temperature,  the  temperature  was  increased  progressively  with  increasing 
acoustic  loading,  generating  accelerometer  and  microphone  data  at  480  C  and 
140  dB,  650°C  and  155  dB,  860°C  and  155  dB,  and  925“C  at  165  dB.  It  was 
clear  from  the  data  at  the  higher  temperatures  and  load  levels  that  the 
full  spectrum  overall  rms  displacement  levels  obtained  by  double  integrat¬ 
ing  the  accelerometer  output  signals  could  not  be  used  to  determine  high 
temperature  strain  levels  due  to  high  amplitude,  low  frequency  displace¬ 
ments  (displacement  being  inversely  proportional  to  frequency  squared  for  a 
given  "g"  level)  that  were  well  below  the  panel  response  frequency  range 
and  therefore  would  not  be  proportional  to  panel  strain  levels.  It  is 
important  to  remember  here  that  since  the  accelerometer  is  mounted  on  the 
panel  fixture,  it  is  measuring  fixture  response,  some  of  which  is  not 
related  to  panel  response. 


After  reviewing  the  various  frequency  spectra,  it  was  decided  to  re-analyze 
the  data  to  generate  overall  rms  levels  over  selected  frequency  bandwidths 
that  would  encompass  a  high  percentage  of  the  full -spectrum  overall  rms 
strains  and  eliminate  the  low  frequency  displacements.  If  a  consistent 
strain-displacement  relationship  could  be  established  at  room  temperature 
within  a  frequency  bandwidth  such  that  the  strains  could  be  related  to  the 
full-spectrum  overall  rms  strains,  and  if  the  same  bandwidth  could  be  used  to 
generate  displacements  at  temperatures  that  were  sufficiently  consistent  to 
relate  to  strain  response,  then  it  would  be  possible  to  at  least  make  a 
reasonable  estimate  of  the  test  temperature  strain  level.  It  was  determined 
that  band-passed  response  data  in  the  300  to  600  Hz  frequency  range  gave 
consistent  strain-displacement  ratios  at  room  temperature.  Double-integrated 
band-passed  accelerometer  outputs  (displacements)  were  consistent  with 
increasing  sound  pressure  levels  at  incrementally  increasing  test  temperatures 
up  to  the  925°C/165  dB  endurance  test  conditions.  Table  3  summarizes  the  high 
temperature  test  panel  results. 


TABLE  3.  HIGH  TEMPERATURE  TEST  PANEL  RESULTS. 


TEST  PANEL 

TEST 

TEMPERATURE 

OVERALL 
SOUND 
PRESSURE 
LEVEL  (dB) 

HIGHEST 
ESTIMATED 
OVERALL  RMS 
STRAIN 

(MICROSTRAIN) 

EXPOSURE  TIME, 
ESTIMATED  FATIGUE 

CYCLES  AND  COMMENTS 

(°F) 

rc) 

BETA  21S  TMC 
PANEL 

1500 

815 

165 

NOT  ESTIMATED 

3  1/2  HRS,  3x10“  CYCLES, 
STIFFENERS  CRACKED  AT 
MID-SPAN 

CARBON-CARBON 
PANEL  NO.  2 

150 

155 

TTTORSTTTMo^TYOlsr 

NO  FAILURE 

155 

219  ^ 

3  1/2  HRS,  2.3x10°  CYCLtS, 
NO  FAILURE 

160 

316 

3  1/2  HRS.  2.3x10“  CYCLES, 
NO  FAILURE 

165 

453 

1  HR,  6.4xl0‘>  cycles! 

CRACKS  AT  STIFFENER  ENDS 

925 

165 

103 

10  HRS,  1.7x10'  CYCLES, 

NO  FAILURE 

941 


It  should  be  noted  that  carbon-carbon  panels  1  and  2  exhibited  cracks  at 
the  stiffener  ends,  whereas  the  maximum  measured  strains  were  at  the  edges 
of  skin  bays.  Consequently,  the  actual  strain  levels  at  the  crack 
locations  were  either  higher  than  the  measured  levels  or  there  were 
significant  stress  concentrations  at  the  stiffener  terminations. 


5.  COMPARISON  OF  ANALYTICAL  AND  TEST  RESULTS  FOR  CARBON-CARBON  PANELS 

MSC  NASTRAN  was  used  to  perform  finite  element  analyses  on  the  three 
carbon-carbon  panels  that  were  subjected  to  the  sonic  fatigue  testing 
described  in  Section  4.  The  oxidation  resistant  coating  was  modeled  as  a 
non-structural  mass,  which  is  compatible  with  the  panel  test  results. ^ 
Natural  frequencies,  mode  shapes  and  acoustically  induced  random  strain 
levels  were  analytically  determined  for  room-temperature  conditions  and 
compared  to  the  room-temperature  panel  test  results. 

Acoustically  induced  random  stresses  were  analytically  determined  on  a 
mode-by-mode  basis  using  the  finite  element  generated  mode  shapes  and  a 
Rohr  computer  code  based  on  an  analytical  procedure  presented  in 
Reference  4.  This  procedure  extends  Miles*  approach  (Reference  5)  to 
include  multi-modal  effects  and  the  spatial  characteristics  of  both  the 
structural  modes  and  the  impinging  sound  field. 

Table  4  shows  the  calculated  and  measured  frequencies,  overall  rms  strain 
levels  and  the  strain  spectrum  levels  for  the  in-phase  stiffener  bending 
mode  for  the  carbon-carbon  panels  at  room  temperature. 


TABLE  4.  CALCULATED  AND  MEASURED  RESPONSE  FREQUENCIES  AND  STRAIN 
LEVELS  FOR  CARBON-CARBON  PANELS  AT  ROOM  TEMPERATURE. 


STRAIN  LEVELS  AT  EDGE  OF  SKIN  BAY 

NATUF 
FREQUE 
OF  IN-F 
MODE 
(Hzl 

\fKL 

:ncy 

>HASE 

OVERALL  RMS  STRAIN 
(MICROSTRAIN) 

STRAIN 

SPECTRUM  LEVEL 
IN-PHASE  MODE 
(MICROSTRAIN/Hz) 

FE  ANALYSIS 

MEASURED 

FE  ANALYSIS 

MEASURED 

FE  ANALYSIS 

MEASURED 

PANEL  1 
a65  dBl 

305 

267 

510 

305 

84 

60 

PANEL  2 
(145  dBl 

190 

155  & 
171 

133 

126 

40 

41 

PANEL  3 
(165  dB) 

460 

423 

77 

69 

16 

16 

942 


The  above  results  show  good  agreement  between  the  finite  element  generated 
values  and  those  measured.  The  level  of  agreement  is  particularly  good  for 
the  strain  spectrum  levels,  which  are  typically  more  difficult  to 
accurately  predict.  Figure  7  shows  the  finite-element  frequency  solution 
for  Panel  3.  The  in-phase  mode  shape  can  be  seen  to  have  an  overall  modal 
characteristic  due  to  the  relatively  low  bending  stiffness  of  the 
stiffeners  for  the  skin  thickness  used.  Figure  8  shows  the  measured  and 
finite-element  generated  strain  frequency  spectra  for  Panel  3. 

Details  of  the  finite-element  analyses  and  models  are  contained  in 
References  1  and  2. 


6.  CONCLUSIONS  AND  RECOMMENDATIONS 

1.  The  high  temperature  testing  techniques  and  strain  measuring 
procedures  successfully  generated  usable  random  fatigue  S-N 
curves  and  panel  response  data.  The  use  of  strain-displacement 
ratios  were  shown  to  be  an  effective  alternative  to  high 
temperature  strain  gauge  measurements. 

2.  In  general,  the  materials  and  structural  concepts  tested 
demonstrated  their  suitability  for  hypersonic  flight  vehicle  skin 
panel  applications.  The  major  exception  was  Titanium-Aluminide 
Super  Alpha  Two  which  was  determined  to  be  too  brittle. 

3.  Inhibited  carbon-carbon  exhibited  significantly  higher  random 
fatigue  strength  at  980°C  than  did  the  uninhibited  carbon-carbon 
—  two  to  three  times  the  random  fatigue  endurance  strain  level. 

4.  Thermally  exposed  enhanced  SiC/SiC  had  comparable  fatigue 
strength  to  that  of  inhibited  carbon-carbon  at  980 °C. 

5.  The  TMC  specimens  usefully  demonstrated  the  fatigue  strength  of 
the  TMC  concept  and  the  need  to  develop  the  concept  to 
incorporate  higher  temperature  capability  titanium  matrix 
materials. 

6.  Titanium  6-2-4-2-Si  exhibited  high  fatigue  strength  in  the  590°C 
to  650 “C  temperature  range  and  also  demonstrated  the  need  for  TMC 
materials  to  utilize  higher  temperature  matrix  materials  in  order 
to  be  cost  effective  against  the  newer  titanium  alloys. 

7.  The  level  of  agreement  between  the  finite  element  analysis 
results  for  the  carbon-carbon  panels  and  the  progressive-wave 
tube  test  data  demonstrated  the  effectiveness  of  the  analytical 
procedure  used.  The  analysis  of  structures  utilizing  materials 
such  as  carbon-carbon  clearly  presents  no  special  difficulties 
providing  the  material  properties  can  be  well  defined. 


943 


8  It  is  recommended  that  further  tests  be  conducted  similar  to 
those  performed  in  this  program  but  with  greater  emphasis  on 
testing  panels  having  dimensional  variations  in  order  to  develop 
design  criteria  and  life  prediction  techniques.  Such  testing 
should  be  performed  on  those  structural  materials  and  design 
concepts  that  emerge  as  the  major  candidates  for  flight  vehicle 
applications  as  materials  development  and  manufacturing 
techniques  progress. 

REFERENCES 


1  R  D.  Blevins  and  I.  Holehouse,  "Thermo-Vibro  Acoustic  Loads  and 
rkigue  of  Hypersonic  Flight  Vehicle  Structure,"  Rohr,  Inc. 
Engineering  Report  RHR  96-008,  February  1996. 

2.  United  States  Air  Force  Systems  Command,  Flight  Dynamics 
Laboratory  Final  Technical  Report,  Contract  No.  F33615-87-C-33^^/, 
to  be  published. 

3.  R.  D.  Blevins,  "Fatigue  Testing  of  Carbon-Carbon  Acoustic  Shaker 
Table  Test  Coupons,"  Rohr,  Inc.  Engineering  Report  RHR  91-087, 
September  1991. 

4.  R.  D.  Blevins,  "An  Approximate  Method  for  Sonic  Fatigue  Analysis 
of  Plates  and  Shells,"  Journal  of  Sound  and  Vibration,  Vol.  129, 
51-71,  1989. 

5.  J.  W.  Miles,  "On  Structural  Fatigue  Under  Random  Loading," 

Journal  of  Aeronautical  Sciences,  Vol.  21,  November  1954. 


944 


A.  Test  Configuration  for 
Material  Coupons 


Fixture 


Specimen 
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B.  Test  Configuration  for 
Carbon-Carbon  Integral 
Stiffener  Specimens 


Accelerometer 


t 


C.  Test  Configuration  For 
■  Carbon-Carbon  Mechanically 
Fastened  Stiffener  and  All 
Titanium  Diffusion  Bonded 
Joint  Specimens 


FIGURE  1  Typical  Strain  Gauge  Locations  and  Test  Configurations 
for  Material  Coupon  and  Joint  Subelement  Shaker  Test 
Specimens 


945 


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FIGURE?  Finite  Element  Solution  for  In-Phase  Overall  Mode  — 
Carbon-Carbon  Panel,  Concept  3 


950 


STRRIN  [Ins  xlB^G/Inl 


952 


SCALING  CONCEPTS  IN  RANDOM  ACOUSTIC  FATIGUE 

BY 

Marty  Ferman*  and  Howard  Wolfe** 

ABSTRACT 

Concepts  are  given  for  scaling  acoustic  fatigue  predictions  for  application  to 
extreme  environmental  levels  based  on  testing  “  scaled”  structures  at 
existing,  lesser  environmental  levels.  This  approach  is  based  on  scaling  a  test 
structure  to  fit  within  the  capabilities  of  an  existing  test  facility  to  attain 
fatigue  results,  and  then  using  analytical  extrapolation  methods  for  predicting 
the  full  scale  case  to  achieve  accurate  design  results.  The  basic  idea  is  to 
utilize  an  existing  acoustic  fatigue  facility  to  test  a  structure  which  has  been 
designed  (scaled)  to  fatigue  within  that  facility’s  limits,  employing  the 
appropriate  structural  properties  ( such  as  thinning  the  skins,  etc.).  Then,  the 
fatigue  life  of  the  actual  structure  is  determined  by  analytically  scaling  the  test 
results  to  apply  to  the  full  scale  case  (  thicker)  at  higher  noise  levels  for 
example.  Examples  are  given  to  illustrate  the  approach  with  limits  suggested, 
and  with  the  recognition  that  more  work  is  needed  to  broaden  the  idea. 

BACKGROUND 

While  it  is  important  to  continually  expand  the  capability  of  acoustic  test 
facilities  ,  it  is  perhaps  equally  important  to  be  able  to  work  with  existing 
facilities  at  any  time.  That  is,  facility  expansions,  enhancements  ,  and 
modernization’s  should  always  be  sought  from  time  to  time,  so  long  as 
practical  and  affordable  from  cost  effective  considerations.  Limits  should  be 
pushed  to  accommodate  larger  sizes  of  test  specimens  with  higher  noise  levels 
with  wider  ranges  of  frequencies,  with  wider  ranges  of  temperatures,  and  with 
better  capabilities  for  applying  pressures  along  with  any  one  of  several  types 
of  preloads.  These  are  costly  considerations  and  require  considerable  time  to 
accomplish.  Facility  rental  can  be  used  in  some  cases  to  bolster  one’s  testing 
facilities,  however  if  the  application  suggests  a  situation  beyond  any  available 
facility  for  the  required  design  proof,  then  an  alternate  is  needed.  Thus  the 
scaling  concept  suggested  here  is  a  viable  and  useable  possibility. 

The  Author’s  basis  for  the  approach  stems  from  their  extensive,  collective, 
experience  in  Structural  Dynamics,  especially  work  in  Acoustic  Fatigue, 
Fluid-Structure  Interaction,  Buffet ,  and  Aeroelasticity/Flutter ,  and 
particularly  from  experience  with  flutter  model  testing,  in  which  it  is  quite 
common  to  ratio  test  results  from  a  model  size  to  full  scale  for  valid 


Assoc.  Prof,  Aerospace  and  Mech.  Engr.  Dept,  Parks  College,  St.  Louis 
Univ.,  Cahokia,  IL,  62206,  USA 

**  Aerospace  Engineer,  Wright  Laboratory,  Wright  Patterson  AFB,  OH, 
45433,  USA 


953 


predictions  .  Flutter  is  a  well  recognized  area  where  model  data  is  commonly 
used  in  nondimensional  form  to  establish  design  margins  of  safety,  as  typified 
in  Ref  (1-2).  Flutter  can  be  nondimensionalized  quite  broadly  as  pointed  out 
in  many  works,  and  is  clearly  done  for  a  wide  range  of  general  cases  using  the 
“so-called”  Simplified  Flutter  Concept,  Ref  (3) .  The  degree  of  the  use  of  the 
flutter  model  scaling  rules  varies  considerably  today,  because  some  people 
are  testing  as  much  or  more  than  ever,  while  others  are  testing  less  and  relying 
more  heavily  on  advanced  theories  such  as  Computational  Fluid  Dynamics, 
CFD.  However,  the  basic  ideas  in  flutter  model  scaling  are  still  POWERFUL! 
In  fact,  this  concept  has  fueled  the  Author’s  desires  to  develop  the 
“acoustical  scaling”  used  in  the  approach  presented.  Moreover,  when  starting 
to  write  this  paper,  the  Authors  realized  that  this  type  of  scaling  is  also 
common  to  many  related  areas  of  structural  dynamics,  and  thus  chose  to 
include  some  examples  of  those  areas  to  emphasize  the  main  point  here! 

For  example,  experience  in  fluid-structure  interaction  and  fatigue  of  fuel  tank 
skins,  a  related  work  area,  serves  as  another  example  of  scaling  structures  to 
demonstrate  accurate  predictions  with  widely  varied  environmental  levels, 
and  a  multitude  of  configurations.  Scaling  and  nondimensional  results  were 
used  extensively  in  Ref  (4  -13),  and  are  cited  here  because  of  the  immense 
data  base  accumulated.  The  work  at  that  time  did  not  necessarily  define 
scaling  as  used  here ,  but  hindsight  now  suggests  that  there  is  a  clear  relation. 

It  is  becoming  well  recognized  that  Buffet  is  easily  scaled  ,  and  many 
engineers  and  investigators  are  now  employing  scaling  of  pressures  from 
model  to  full  size  applications,  and  are  also  using  scaled  model  response  to 
predict  full  scale  cases  .  Some  of  the  earliest  and  some  of  the  more  modern 
results  clearly  show  this  aspect.  For  example.  Ref  (14  -16)  are  typical,  quite 
convincing,  and  pace  setting  regarding  scaled  data.  Buffet  models  which  are 
much  more  frail  that  the  full-scale  cases  are  used  to  develop  data  for  full  scale 
applications,  and  besides  giving  full  sized  results,  provide  a  guide  to  safe 
flight  testing  as  has  been  done  more  extensively  with  flutter  testing. 

Obviously,  acoustical  response  and  fatigue  phenomenon  are  also 
nondimensionalizable  and  scaleable.  Ref  (17-18),  for  example.  This  point  is 
being  taken  further  here;  that  is  ,  scaling  will  be  used  to  take  better  advantage 
of  limited  facility  testing  capability  to  predict  more  severe  situations,  as  is 
used  in  the  case  of  flutter  model  testing  where  a  larger  specimen  is  predicted 
from  tests  of  a  smaller  structure  using  similarity  rules.  Here  in  the  acoustic 
application,  a  thinner  ,  or  otherwise  more  responsive  specimen,  is  tested  and 
then  analytical  means  are  used  to  make  the  prediction  for  the  nominal  case. 

APPROACH 

The  method  is  shown  here  is  basically  an  extension  of  the  flutter  model 
scaling  idea,  as  applied  to  acoustical  fatigue  testing  with  a  particular  emphasis 
on  random  applications.  The  technique  will  also  work  for  sine  type  testing  in 


954 


acoustical  fatigue,  and  perhaps  it  will  be  even  more  accurate  there,  but  most  of 
today’s  applications  are  with  random  testing,  notably  in  the  aircraft  field. 
Thus  it  is  in  this  area  where  the  method  should  find  more  application  .  The 
Authors  have  a  combined  professional  work  experience  of  some  70+  years 
and  thus  have  tried  to  focus  this  extensive  background  on  an  area  where  gains 
can  be  made  to  help  reduce  some  costs  while  making  successful  designs,  by 
using  lesser  testing  capability  than  might  be  more  ideally  used.  It  is  believed 
that  the  best  testing  for  random  acoustic  fatigue,  is  of  course,  with  (a)  the 
most  highly  representative  structure,  and  as  large  a  piece  as  can  be  tested, 
both  practically  and  economically,  (b)  the  most  representative  environmental 
levels  in  both  spectrum  shape  and  frequency  content,  (c)  test  times  to 
represent  true  or  scaled  time,  as  commonly  accepted,  (d)  temperatures  should 
be  applied  both  statically  and  dynamically,  and  finally  (e)  preloading  from 
pressures,  vibration,  and  from  boundary  loading  of  adjacent  structure. 
Frequently,  testing  is  done  to  accomplish  some  goal  using  a  portion  of  these 
factors,  and  the  remainder  is  estimated  .  Thus  the  Authors  believed  that  there 
is  a  high  potential  to  extend  the  flutter  model  approach  to  acoustical 
applications. 

Recall  that  in  the  flutter  model  approach  ,  the  full  scale  flutter  speed  is 
predicted  by  the  rule 


((Vf)a)p-  [  ((Vi.)m)e/  ((  Vf)m)c]  X  [  ((Vf)a)c]  (^) 

where  Vp  is  flutter  speed,  the  subscripts  M  and  A  refer  to  model  and  aircraft 
respectively,  the  subscript  C  refers  to  calculated,  and  the  subscript  P  refers  to 
predicted.  Thus  the  equation  suggests  that  the  full  scale  predicted  flutter  speed 
is  obtained  by  taking  the  ratio  of  experimental  to  calculated  flutter  speed  for 
the  model  and  then  multiplying  by  a  calculated  speed  for  the  airplane.  These 
flutter  model  scaling  ideas  are  covered  in  any  number  of  References,  i.e.  Ref 
(lo),  for  example. 

The  same  concept  can  be  utilized  in  acoustic  fatigue,  i.e.  the  strain  at  fatigue 
failure  relation,  (8,N)  can  be  scaled  from  model  structure  tested  at  one  level 
and  then  adjusted  for  structural  sizing  and  environmental  levels.  This  relation 
can  be  addressed  as  done  for  the  flutter  case: 

((e,N)a)p=[((  s,N)m)e  /  ((e,N)m)c]  X  [((e.N)a)c]  (2) 

where  s  is  strain,  and  N  is  the  number  of  cycles  at  failure,  where  as  above  in 
Eq  (1) ,  the  subscripts  M  and  A  refer  respectively  to  Model  and  Full  Scale  for 
parallelism,  while  the  subscripts  E,  C,  and  P  have  the  same  connotation  again, 
namely,  experimental,  calculated,  and  predicted.  Thus  the  full  scale  case  is 


955 


predicted  from  a  subscale  case  by  using  the  ratio  of  experimental  to 
theoretical  model  results  as  adjusted  by  a  full  scale  calculation.  Flutter  model 
scaling  depends  upon  matching  several  nondimensional  parameters  to  allow 
the  scaling  steps  to  be  valid.  While  these  same  parameters  are,  of  course,  not 
necessarily  valid  for  the  acoustic  relationships,  other  parameters  unique  to 
this  acoustical  application  must  be  considered,  and  will  be  discussed. 
Accurate  predictions  for  the  method  relies  on  extensive  experience  with  the 
topic  of  Acoustic  Fatigue  in  general,  because  concern  is  usually  directed 
towards  the  thinner  structure  such  as;  panels,  panels  and  stiffeners,  and  panels 
and  frames,  bays  (a  group  of  panels),  or  other  sub-structure  supporting  the 
panels.  These  structures  are  difficult  to  predict  and  are  quite  sensitive  to  edge 
conditions,  fastening  methods,  damping,  combination  of  static  and  dynamic 
loading,  and  temperature  effects.  Panel  response  prediction  is  difficult ,  and 
the  fatigue  properties  of  the  basic  material  in  the  presence  of  these  complex 
loadings  is  difficult.  However,  the  experienced  Acoustic  Fatigue  Engineer  is 
aware  of  the  limits,  and  nonnally  accounts  for  these  concerns.  Thus  the 
method  here  will  show  that  these  same  concerns  can  be  accounted  for  with  the 
scaling  approach  through  careful  considerations. 

The  Authors  believe  that  the  method  is  best  explained  by  reviewing  the 
standard  approach  to  acoustic  fatigue,  especially  when  facility  limits  are  of 
major  concern.  Fig  (1)  was  prepared  to  illustrate  these  points  of  that 
approach.  Here  it  is  seen  that  key  panels  for  detail  design  are  selected  from  a 
configuration  where  the  combination  of  the  largest,  thinnest,  and  most 
severely  loaded  panels  at  the  worst  temperature  extremes  and  exposure  times 
are  considered.  These  can  be  selected  by  many  means  ranging  from  empirical 
methods,  computational  means,  and  the  various  Government  guides.  Ref  (17- 
18),  for  example.  Then  detailed  vibration  studies  are  run  using  Finite 
elements  ,  Rayleigh  methods.  Finite  Difference  methods,  etc.  to  determine  the 
modal  frequencies  and  shapes,  and  frequently  linearity  is  assessed.  Then 
acoustical  strain  response  of  the  structure  is  determined  for  sine,  narrowband, 
and  broadband  random  input  to  assess  fatigue  life  based  on  environmental 
exposure  times  in  an  aircraft  lifetime  of  usage.  These  theoretical  studies  are 
then  followed  by  tests  of  the  worst  cases,  where  vibration  tests  are  conducted 
to  verify  modal  frequencies,  shapes,  and  damping,  and  linearity  is  checked 
again  for  the  principal  modes.  This  is  followed  by  acoustical  strain  response 
tests  where  the  strain  growth  versus  noise  levels  is  checked,  again  employing 
sine,  narrowband  and  broadband  random  excitation.  Note  the  figure  suggests 
that  data  from  the  vibration  tests  are  fed  back  to  the  theoretical  arena  where 
measured  data  are  used  to  update  studies  and  to  correlate  with  predictions, 
especially  the  effect  of  damping  on  response  and  fatigue,  and  of  course,  the 
representation  of  nonlinearity.  Also,  the  measured  strain  response  is  again 
used  to  update  fatigue  predictions.  These  updates  to  theory  are  made  before 
the  fatigue  tests  are  run  to  insure  that  nothing  is  missed.  However,  in  this 


956 


case,  the  required  sound  presssure  level  SPL  in  (dB)  is  assumed  to  exceed  the 
test  chamber’s  capability.  Thus  ,  as  shown  in  the  sketch  in  Fig  (2)  the  key 
strain  response  curve,  s  vs  dB,  is  extrapolated  to  the  required  dB  level.  This 
data  is  merged  with  the  strain-to-failure  curve  at  the  right  to  establish  the  cycle 
count,  N,  giving  the  (s,  N)  point  for  this  case.  The  extrapolated  data  provides 
some  measure  of  the  estimated  life,  but  again  is  heavily  dependent  upon  the 
accuracy  of  the  basic  strain  response  curve,  and  is  especially  dependent  on 
whether  high  confidence  exists  at  the  higher  strains.  Linear  theory  is  also 
shown  in  this  case,  indicating  it  overpredicts  the  test  strain  response  and  hence 
shows  a  shortened  fatigue  life  compared  to  test  data,  as  is  generally  the  case 
in  today’s  extreme  noise  levels.  This  illustration  is  highly  simplified, 
because  experienced  designers  readily  know  that  it  is  difficult  to  predict  even 
simple  panels  accurately  at  all  times,  let  alone  complex  and  built-up  structure 
consisting  of  bays  (multi-panel);  this  will  addressed  again  later  in  the  paper. 

The  new  concept  of  scaled  acoustic  fatigue  structures  is  shown  on  the  sketch 
of  Fig  (3)  where  the  standard  method  is  again  shown,  but  smaller  in  size,  to 
refocus  the  thrust  of  the  new  idea.  The  scaling  process  parallels  the  standard 
approach,  and  actually  complements  that  approach,  so  that  the  two  can  be  run 
simultaneously  to  save  time,  costs,  and  manpower.  Here  the  panel  selection 
process  recognizes  that  the  design  application  requires  environments  far  in 
excess  of  available  facility  capability.  Thus  the  scaling  is  invoked  in  the 
beginning  of  the  design  cycle.  As  the  nominal  panels  (bays)  are  selected  and 
analyzed  for  vibration,  response  and  fatigue,  scaled  structures  are  defined  to 
provide  better  response  within  the  existing  chamber  ranges  so  that  they  can  be 
fatigued  and  then  the  results  can  be  rescaled  to  the  nominal  case.  In  this 
manner,  appropriate  designs  can  be  established  to  meet  safety  margins  with 
more  confidence,  and  will  avoid  costly  redesign  and  retrofitting  at 
downstream  stages  where  added  costs  can  occur  and  where  down  times  are 
difficult  to  tolerate.  The  concept  is  further  illustrated  in  the  sketch  of  Fig  (4). 
Here  the  strain  response  curve  of  the  nominal  case  and  that  of  the  scaled 
version  are  combined  with  strain  to  failure  data  (  coupon  tests)  to  show  fatigue 
results.  Note  the  strain  response  for  the  nominal  case  at  the  highest  dB  level 
available  gives  the  fatigue  value  at  point  A,  while  the  extrapolated  data  for 
this  curve  gives  point  B.  The  scaled  model  being  more  responsive  gives  the 
point  C,  and  when  rescaled  gives  the  point  D  which  differs  slightly  from  the 
extrapolated  point  B  as  it  most  likely  will,  realistically.  More  faith  should  be 
placed  on  data  from  an  actual  fatigue  point  than  a  point  based  on  the  projected 
strain  response  curve.  Note,  Fig  (5)  illustrates  the  winning  virtue  of  the 
scaleable  design.  The  figure  shows  a  hypothetical  set  of  test  data  for  the  (s  ,N) 
for  a  structure  for  various  SPLs  for  the  nominal  case,  open  circles,  and  for  the 
fatigue  results  of  the  scaled  model  ,  closed  squares.  The  scaled  model  was 
assumed  to  be  thinner  here  for  example,  and  that  the  scaled  data  is  also 


957 


rescaled  to  fit  the  nominal  curve  here.  The  most  interesting  aspect  is  shown 
by  the  two  clusters  of  data ,  denoted  as  A  and  B  where  there  are  rough  circles 
about  drawn  about  the  clusters.  Here  the  emphasis  is  that  tests  of  the  scaled 
model  (  and  rescaled  data )  are  used  to  find  the  higher  strain  conditions  which 
cannot  be  found  from  the  nominal  case.  In  both  cases  at  the  highest  strain 
levels,  the  facility  is  used  to  its  limits,  but  with  enough  testing  with  the  thinner 
case,  adequate  data  is  available  to  make  the  prediction  more  accurate  using 
Eq  (2)  for  the  final  correlation  as  shown  here.  The  statistical  scattering  of  the 
scaled  data  will  be  an  accurate  measure  for  the  nominal  case,  particularly 
when  compared  to  estimates  based  on  extrapolation  of  the  strain  response  for 
the  nominal  case.  There  are  many  cautions  to  be  noted  with  this  approach  as 
there  are  with  all  acoustic  fatigue  methods,  and  of  course,  tests.  First,  the 
linearity  of  the  modes,  either  in  unimodal  sine  excitation,  multi-mode  sine, 
narrow  band  or  broadband  random  must  be  carefully  handled.  The  strain 
response  of  individual  locations  throughout  the  structure  must  be  carefully 
monitored  in  calculations  and  tests  so  that  strain  response  is  truly  understood 
and  used  to  define  fatigue  life  carefully.  This  is  difficult  to  do  in  many 
applications  where  widely  varying  conditions  and  durations  require  some  type 
of  Miner  Rule  combination  to  provide  a  true  measure  of  fatigue.  Similarly, 
strain  risers  at  fasteners,  discontinuities,  holes,  frames,  stiffeners,  material 
changes  along  with  temperature  gradients,  temperature  transients,  require 
final  “tweeks”  to  predictions,  regardless.  Nonlinearity ,  especially  in  the 
multi  -mode  case,  is  one  of  the  most  formidable  foes  to  conquer  for  any 
application. 


APPLICATIONS  AND  EXAMPLES 

The  tests  of  an  Aluminum  panel  of  size  10x20  in.  and  with  a  thickness  of 
0.063  thickness  ,  Ref  (19),  will  be  used  to  illustrate  the  technique.  The  panel 
has  approximately  fixed-fixed  edge  conditions,  and  is  quite  nonlinear  in 
terms  of  strain  response.  Ref  (19).  The  measured  strain  response  for  the  panel 
is  shown  in  Fig  (6)  along  with  an  estimated  response  curve  for  a  thinner  panel 
(  0.040  in  )  based  on  the  test  data.  For  this  case  it  is  assumed  that  data  were 
needed  at  175  dB,  while  the  facility  could  only  achieve  164  dB.  The  strain 
response  for  the  thinner  case  was  estimated  using  the  classical  equation  for 
the  amplitude  of  response,  5  ; 


\\fdxdy\pSD,{f)] 


(3) 


where  M  is  the  generalized  mass,  co  is  the  natural  frequency,  §  is  the  viscous 
damping  factor  ,  (j)  is  mode  shape,  PSDp  is  the  pressure  Power  Spectral 
Density,  and  x,y  are  the  positional  coordinates  along  the  plate.  Since  strain  , 
s,  is  proportional  to  the  amplitude. 


958 


s=  (t/2)(3(ti^/ax^)S 


(4) 

Combining  Eq  (3  and  4)  shows  that  the  strain  response  curves  are 
proportional  to  the  thickness  factor,  as  given  by: 

s  2  ~  (  ti  /  t2)  (5) 

However,  it  must  be  noted  that  this  case  is  nonlinear,  and  thus,  this  result  is 
not  exactly  correct,  but  simply  used  for  an  illustration  here.  The  actual  data  for 
the  0.063  thickness  is  extrapolated  to  a  required  175  dB,  showing  a  strain  of 
1000  micro  in/in.  The  estimated  curve  for  the  thickness  of  0.040  in.  shows  ,of 
course,  a  greater  response  at  all  dB  levels  as  it  should,  and  moreover  shows 
that  only  150  dB  are  needed  to  achieve  the  1000  micro  -  strain  condition. 
Moreover,  the  thinner  panel  will  exhibit  large  enough  strains  at  the  lower 
SPLs  to  improve  the  fatigue  curve  where  the  thicker  panel  is  insensitive. 
Taking  the  example  a  step  further,  the  fatigue  point  of  the  nominal  case  is 
shov^Ti  on  a  strain  to  failure  plot  in  Fig  (7) ,  employing  beam  coupon  tests  of 
Ref  (11),  which  were  shown  to  be  excellent  correlators  with  panel  fatigue  in 
the  collection  of  work  in  Ref  (6-13).  The  fatigue  point  for  the  164  dB 
excitation,  800  micro-strain,  is  shown  as  a  triangle,  while  the  extrapolated 
data  for  175  dB  is  shown  as  the  flagged  triangle.  One  test  point  exists  for  the 
nominal  case.  Ref  (20) ,  and  is  shown  by  a  star  symbol.  Data  for  the  thinner 
panel  are  shown  as  circles  at  the  various  strain  to  cycle  count  cases  for  the 
various  SPLs  corresponding  to  the  beam  curve.  Notably,  these  points  can  be 
seen  to  produce  shorter  fatigue  cases  as  they  should  due  to  increased  strains, 
but  note  that  they  are  also  at  lower  frequencies  which  would  give  a  longer  test 
time  than  if  they  were  the  nominal  thickness.  The  scaled  model  is  seen  to 
produce  the  same  point  as  the  extrapolated  case  in  this  hypothetical  case  for 
the  1000  microstrain  case  (  again,  at  two  different  dB  levels  for  two 
thicknesses).  A  SPL  of  150  dB,  rather  benign,  is  seen  to  be  quite  effective. 
The  actual  fatigue  point  at  164  dB  for  the  nominal  case  required  3  hours  and 
was  predicted  to  be  2.8  hrs.  The  estimated  fatigue  for  the  extrapolated  case  of 
175  dB  was  estimated  to  be  1.7  hrs,  while  the  scaled  point  from  the  thinner 
panel  was  estimated  to  be  2.2  hours  which  is  slightly  off,  but  the  Authors 
have  had  to  rely  on  log  plots  for  much  of  the  data  and  thus  lack  someaccuracy. 
Because  of  lack  of  actual  data ,  the  scatter  from  the  estimate  vs  the  test  of  the 
nominal  case  was  used  to  scatter  the  estimate  for  the  1000  microstrain  case, 
flagged  dark  circle,  as  if  the  use  of  Eq  (2)  had  been  employed  directly.  One 
must  be  careful  here,  because  there  can  be  a  vast  difference  between  theory 
and  test,  and  this  can  mislead  inexperienced  persons  applying  these  methods. 
As  noted  earlier,  related  work  in  fluid-structure  and  buffet  ,  actually 
demonstrate  this  type  of  scaling.  To  illustrate,  several  figures  are  republished 


959 


here  to  make  this  point  rather  clear.  Fig  (8)  of  Ref  (8  )  shows  the  dynamic 
bending  strains  in  the  bottom  panel  of  an  otherwise  rigid  fuel  tank  which  is 
being  excited  vertically  with  moving  base  input.  The  vertical  axis  is  strain 
while  the  abscissa  is  the  number  of  g’s  input.  Three  panel  thicknesses  and  four 
depths  of  fluid  (water  in  this  case)  were  used.  Note  the  sharp  nonlinear  effect 
in  the  response,  rather  than  linear  response  growth  as  force  increases. 
Interestingly,  the  data  was  nondimensionalized  into  the  curve  of  Fig  (9),  Ref 
(8),  which  was  originally  intended  for  a  design  chart  to  aid  in  developing 
strain  response  characteristics  for  use  in  fatigue.  This  curve  displays  a 
parameter  of  response  as  the  ordinate  vs  an  excitation  parameter  on  the 
abscissa.  Here,  E  is  Young’s  modulus,  p  is  density,  t  is  thickness,  a  is  the 
panel  length  of  the  short  side,  h  is  fluid  depth,  and  the  subscripts,  p  and  F  refer 
to  panel  and  fluid,  respectively.  A  point  not  realized  previously  is  that  the 
scaling  shows  that  the  thinner  case  can  be  used  to  represent  the  thicker  panel 
under  the  appropriate  conditions  and  when  nonlinearity  is  carefully 
considered.  More  data  with  the  thinner  panels  at  the  extreme  conditions  were 
unfortunately  not  taken  in  several  cases  of  strain  response  because  of  concern 
with  accumulating  too  many  cycles  before  running  the  actual  fatigue  tests; 
else  the  thinner  cases  could  have  shown  even  more  dramatically  the  scale 
effect. 

Buffet  has  been  of  more  interest  in  the  past  15  years  because  of  high  angle  of 
attack  operation  of  several  modem  USAF  fighters.  Much  effort  was  placed 
upon  research  with  accurately  scaled  models  to  detennine  if  these  could  be 
employed  as  in  prior  flutter  work.  The  answer  was  YES!  Several  figures  were 
taken  from  Ref  (16)  to  illustrate  scaling  of  data  from  a  model  of ,  a  fraction 
of  the  size  of  a  fighter,  to  the  full  scale  quantity.  Fig  (10)  shows  the  correlation 
between  scaled-up  model  data,  flight  test,  and  two  sets  of  calculations  over  a 
wide  range  of  aircraft  angle  of  attack  for  the  F/A-18  stabilator.  The  data  is  for 
inboard  bending  and  torsion  moment  coefficients  produced  by  buffeting 
loads.  The  scaled  model  data  correlates  well,  the  calculations  using  Doublet 
Lattice  (DLM)  aerodynamics  is  close,  while  the  strip  theory  is  not  as  accurate. 
Fig  (11)  shows  similar  type  of  data  for  the  F/A-18  Vertical  Tail  for  outboard 
bending  moment  coefficients.  Here  a  wider  range  of  angle  of  attack  was 
considered,  and  again  scaled  model  data  and  calculations  are  close  to  aircraft 
values.  Both  cases  suggest  that  model  data  can  be  used  to  supplement  full 
scale  work  and  that  when  combined  with  theory  ,  are  a  powerful  aid  to  full 
scale  analysis  and  tests.  These  tests  can  be  used  early  in  the  aircraft  design 
cycle  to  insure  full  scale  success. 

CONCLUSIONS  AND  RECOMMENDATIONS 
An  attempt  was  made  to  employ  a  view  of  acoustical  scaling  different  from 
that  usually  taken.  The  idea  is  to  develop  data  for  a  model  that  fits  within  a 
test  facility’s  capability  and  then  by  using  analytical  methods,  adjust  these 
results  to  the  nominal  case  using  factors  from  the  test  based  on  the  ratio  of 


960 


experimental  to  calculated  data.  This  is  analogous  to  the  flutter  model 
approach.  One  example  is  offered,  and  similar  results  from  related  scaling  in 
fluid  -structure  and  buffet  work  were  shown  to  further  the  point.  While  more 
work  is  needed  to  fully  display  the  concept,  enough  has  been  done  to  inspire 
others  to  dig-in  and  more  fully  evaluate  the  approach.  The  Writers  intend  to 
do  more  research,  since  they  fully  appreciate  this  difficult  task. 

REFERENCES 

1.  Scanlon  ,  R.H.,  and  Rosenbaum,  R.,  “  Introduction  to  The  Study  of  Aircraft 
Vibration  and  Flutter”,  The  MacMillian  Company,  New  York,  1951 

2.  Bisplinghoff,  R.L.,  Ashley  ,  H.  and  ,  Halfman,  R.  L.,  “Aeroelasticity”, 
Addision-Wesley  Publishing  Co.,  NY,  Nov.  1955,  pp.  695-787 

3.  Ferman,  M.  A.,  “Conceptual  Flutter  Analysis  Techniques  -  Final  Report” 
Navy  BuWeps  Contract  NO  w  64-0298-c,  McDonnell  Report  F322,  10  Feb. 
1967 

4.  Ferman,  M.A.  and  Unger,  W.  H.,  “Fluid-Structure  Interaction  Dynamics  in 
Fuel  Cells”,  17th  Aerospace  Sciences  Meeting,  New  Orleans,  La.  Jan  1979 

5.  Ferman,  M.A.  and  Unger,  W.  H.  ,  “Fluid-Structure  Interaction  Dynamics  in 
Aircraft  Fuel  Cells”,  AlAA  Journal  of  Aircraft,  Dec.  1979 

6.  Ferman,  M.A.  ,  et  al  ,  “  Fuel  Tank  Durability  with  Fluid-Structure 
Interaction  Dynamics  ,”  USAF  AFWAL  TR-83-3066,  Sept.  1982 

7.  Ferman,  M..A.,  Unger,  W.  H.,  Saff,  C.R.,  and  Richardson,  M.D.,  “A  New 
Approach  to  Durability  Predictions  For  Fuel  Tank  Skins”  ,  26th  SDM, 
Orlando,  FL,  15-17  April  1985 

8.  Ferman,  M.  A.  ,  Unger,  W.  H.,  Saff,  C.R.,  and  Richardson,  M.D.  ,  “  A 
New  Approach  to  Durability  Prediction  For  Fuel  Tank  Skins”,  Journal  of 
Aircraft,  Vol  23,  No.  5,  May  1986 

9.  Saff,  C.R.,  and  Ferman,  M.A,  “Fatigue  Life  Analysis  of  Fuel  Tank  Skins 
Under  Combined  Loads”,  ASTM  Symposium  of  Fracture  Mechanics  , 
Charleston,  SC,  21  March  1985 

10.  Ferman,  M.A.,  Healey,  M.D.,  Unger,  W.H.,  and  Richardson,  M.D., 
“Durability  Prediction  of  Parallel  Fuel  Tank  Skins  with  Fluid-Structure 
Interaction  Dynamics”,  27th  SDM,  San  Antonio,  TX,  19-21  May  1986 

11.  Ferman,  M.A,  and  Healey,  M.D.,  “Analysis  of  Fuel  Tank  Dynamics  for 
Complex  Configurations,  AFWAL  TR  -87-3066,  Wright-Patterson  AFB,  OH, 
Nov  1987 

12.  Ferman,  M.A.,  Healey,  M.D.  and  Richardson,  M.D.,”  Durability  Prediction 
of  Complex  Panels  With  Fluid-Structure  Interaction”,  29th  SDM, 
Williamsburg,  VA,  18-20  April  1988 

13.  Ferman,  M.A.,  Healey,  M.D.,  and  Richardson,  M.D.,  “A  Dynamicisf  s 
View  of  Fuel  Tank  Skin  Durability,  AGARD/NATO  68th  SMP,  Ottawa, 
Canada,  23-28  April  1989 

14.  Zimmerman,  N.H.  and  Ferman,  M.A.,  “Prediction  of  Tail  Buffet  Loads 
for  Design  Applications,  USN  Report,  NADC  88043-30,  July  1987 


961 


15.  Zimmerman,  N.H.,  Ferman,  M.A.,  Yurkovich,  R.N,  “Prediction  of  Tail 
Buffet  Loads  For  Design  Applications”,  30th  SDM,  Mobil,  AL  ,  3-5  April 
1989 

16.  Ferman,  M.A.,  Patel,  S.R.,  Zimmemian,  N.H.,  and  Gerstemkom,  G.,  “  A 
Unified  Approach  To  Buffet  Response  Response  of  Fighters”, AGARD/NATO 
70th  SMP,  Sorrento,  Italy,  2-4  April  1990 

17.  Cote,  M.J.  et  al,  “Structural  Design  for  Acoustic  Fatigue”,  USAF  ASD- 
TR-63-820,  Oct  1963 

18.  Rudder,  F.F.,  and  Plumblee,  H.E.,  Sonic  Fatigue  Guide  for  Military 
Aircraft”  AFFDL-TR-74-1 12,  Wright-Patterson  AFB,  OH,  May  1975 

19.  Jacobs,  J.H.,  and  Ferman,  M.A.  ,  Acoustic  Fatigue  Characteristics  of 
Advanced  Materials  and  Structures,  “  AGARD/NATO  SMP  ,  Lillehammer, 
Norway,  4-6  May  1994 

20.  McDonnell  Douglas  Lab  Report,  Tech.  Memo  253.4415,  Acoustic  Fatigue 
Tests  of  Four  Aluminum  Panels,  Two  With  Polyurethene  Sprayon”,  27  June 
1984 


Figure  1  -  Standard  Acoustic  Fatigue  Design  Method 


962 


MICROSTRAIN  - 


HYPOTHETICAL  EXAMPLE 


SOUND  PRESSURE  LEVEL  -dB  N- CYCLES  TO  FAILURE 


Figure  2  -  General  Method  for  Estimating  Fatigue  Life  at  SPL 
above  Test  Facility  Capability 


Figure  3  -  Scaling  Method  Fits-in  with  General  Design 
Cycle  for  Acoustic  Fatigue 


963 


SOUND  PRESSURE  LEVEL  -  dB 


N@  cIBreq  ,  Nom.  Ext. 
N  @  dBj^ ,  Re-scaled 
N  @  dBj^ ,  Scaled  Model 
N  -  CYCLES  TO  FAILURE 


Figure  4  -  Hypothetical  Examples  of  Acoustic  Scaling 
to  Tests  at  Higher  SPLs 


HYPOTHETICAL  CASE 


Figure  5  -  Statistical  Aspects  of  Scaling 


MICROSTRAIN  - 


^  MDC  Tests 
Ref.  (19) 


Figure  6  -  Strain  Response  of  an  Aluminum  Panel 

(10  X  20  X  0.063  in.  7075  T6)  Narrowband  Random 


Figure  7  -  Example  of  scaled  Model  of  the  10  x  20  in. 
Aluminum  Panel 


965 


Symbol  Panel  Fluid 
Thickness  Depth 

(in.)  (in.) 

□ 

0.032 

11.0 

0 

0.040 

11.0 

o 

0.063 

11.0 

0 

0.032 

8.0 

A 

0.040 

8.0 

o 

0.063 

8.0 

0 

0.032 

4.0 

0.040 

4.0 

• 

0.063 

4.0 

Figure  8  -  Dynamic  Strain  vs  Excitation  Level  -  Sine 


Figure  9  -  Dynamic  Strain  Parameter  vs  Input  Parameter 

Mean  Strain  at  Fluid  Depth 


966 


0.04 


Inboard  Bonding 


Inboard  Torsion 


0.032 

RMS  Buffet  0.024. 

Moment 
Coeffidant, 

0.01  e 

0.008 

0  . 

0  4  8  12  16  20  24 

Angle  of  Attack  -  degrees 


A 

C 

1 

1 

r 

1  0 

< 

) 

• 

0 

_ ^ 

— 

Right  t» 
Scaladt 
Calculal 
italculat 

— 

St 

ipmods 

0d(str^ 

ed(OUl 

Itast 

theory) 

) 

— 

A 

L 

i 

> 

(j 

I  o 

( 

0 

) 

0  4  8  12  16  20  24 


Angie  of  Attack  -  degrees 


Figure  10  -  F/A-18  Stabilator  Buffet  Correlation  Study 


0.048 

0.04 

0.032 

RMS 
Moment 

CoefficiantO.024 
BM/(QL3) 

0.016 

0.008. 

0 

16  24  32  40  48  56 

Angle  of  Attack  -  degrees 

Figure  11  -  F- 18  Vertical  Tail  Buffet  Response  Moment  Coefficients  for 
Angle-of-attack  Variations 


Outboard  Bending  Moment  (70%  Span,  45%  Chord) 

j— I  j  j 


— 

□  Ca 
OMC 
A  FI 

— 

Liculatsd 

ideiscaX 

gniTastf 

— 

d4ip(MS 

517 

— 

WTTeet 

— 

51.  Run 

— 

56) 

( 

A 

1  ( 

'  *, 

)  < 

) 

" 

J 

h 

*  ( 

> 

( 

I 

1 

t 

1 

Zi 

1 

967 


968 


ACOUSTIC  FATIGUE  II 


THE  DEVELOPMENT  AND  EVALUATION  OF  A  NEW 
MULTIMODAL  ACOUSTIC  FATIGUE  DAMAGE  MODEL 


Howard  R  Wolfe 
WL/FIBG  Bidg24C 
2145  Fifth  St  Ste2 
Astronautics 

Wright-Patterson  AFB,  OH 
45433-7006 
USA 


Robert  G.  White 
Head  of  Department 
Department  of  Aeronautics  and 

University  of  Southampton 
Southampton,  S017 IBJ 
UK 


ABSTRACT 

A  multimodal  fatigue  model  has  been  developed  for  flat  beams  and  plates. 
The  model  was  compared  with  experimental  bending  resonant  fatigue 
lifetimes  under  random  loading.  The  method  was  accurate  in  predicting 
cantilevered  beam  fatigue  lifetimes,  but  under  predicted  clamped-  clamped 
beam  test  results.  For  the  clamped  plate  tests,  one  calculation  was  accurate 
and  the  other  predicted  about  half  the  test  lifetimes.  The  comparisons  and 
the  parameters  affecting  them  are  presented. 

INTRODUCTION 

While  the  single  mode  acoustic  fatigue  theory  is  satisfactory  for  sound 
pressure  levels  around  158  dB  overall  and  below,  there  is  evidence  in  the 
literature  [1,  2,  and  3]  that  above  this  level  the  accuracy  of  the  simple 
response  prediction  method  decreases  with  increasing  sound  pressure 
levels.  The  purpose  of  the  paper  is  two  fold,  first  to  develop  a  multimodal 
acoustic  fatigue  life  prediction  model  ^d  secondly,  to  evaluate  its  accuracy 
in  estimating  the  fatigue  life  theoretically  by  comparing  predictions  with 
experimental  results. 

FATIGUE  MODEL  DEVELOPMENT 

Many  fatigue  models  are  found  in  the  literature.  The  Miner  single  mode 
model  used  by  Bennouna  and  White  [4]  and  Rudder  and  Plumblee  [5]  was 
selected  to  develop  a  multimodal  nonlinear  model.  The  fundamental 
formulation  is  given  by, 

N.-fs^r  «) 


969 


where  N,.  is  the  total  number  of  cycles  to  failure,  P(e)  is  the  peak  strain 
probability  density.  N  is  the  total  number  of  cycles  to  failure  at  incremental 
constant  amplitude  strain  levels  derived  from  a  sinusoidal  strain  versus 
cycles  to  failure  curve.  To  calculate  the  fatigue  life  in  hour,  Eq  (1)  can  be 
expressed  as, 

t  (hours) = 


where  t  is  time,  Pp  (sd)  is  the  peak  standard  deviation  probability  density, 
Nc  is  the  total  number  of  cycles  to  failure  at  a  specified  strain  level  and  ^  is 
the  cyclic  frequency.  When  the  mean  value  is  not  zero,  which  is  the  case 
with  axial  strain  in  the  beam  or  plate,  the  rms  value  is  not  the  standard 
deviation.  The  standard  deviation  is  usually  employed  to  compute  the  time 
to  failure.  Most  of  the  S-N  curves  or  e  -N  curves  are  approximated  as  a 
straight  line  on  a  logarithmic  graph.  The  relationship  between  the  surface 
strain  and  the  cycles  to  failure  is  then. 


8  = 


(3) 


where  K  is  a  constant  and  a  the  slope  of  a  straight  line  on  a  log-log  graph. 
The  cyclic  frequency  4  for  a  single  mode  case  is  taken  as  the  frequency  of 
the  associated  resonance.  Two  types  of  peak  probability  density  techniques 
were  investigated  from  multimodal  nonlinear  strain  responses  [3].  These 
were  called  major  peaks  and  minor  peaks.  The  major  peaks  were  counted 
for  the  largest  peaks  between  zero  crossings.  The  minor  peaks  were 
counted  for  all  stress  reversals  or  a  positive  slope  in  the  time  history 
followed  by  any  negative  slope.  The  effective  cyclic  frequency  is  much 
lower  for  the  major  peak  count  than  the  minor  peak  count.  However,  the 
peak  probability  density  functions  or  PPDFs  compared  for  these  two  cases 
were  almost  the  same.  The  major  peak  method  was  selected  for  further 
study. 

Given  a  particular  peak  probability  density  curve  from  a  measured 
response  in  an  experiment,  the  number  of  peaks  and  the  sampling  time  tg 
can  be  used  to  determine  the  effective  multimodal  cyclic  frequency, 

fcm  =  number  of  peaks  /  tg  (4) 


970 


where  is  the  effective  multimodal  cyclic  frequency.  Substituting  the 
multimodal  cyclic  frequency  into  Eq  (4), 


t  (hours)  = 


Pp(sd) 

K/(e)‘/“ 


3600xfcm 


(5) 


This  model  accounts  for  the  effects  of  axial  strains  which  cause  the  mean 
value  not  to  be  zero,  nonlinear  response  and  multimodal  effects.  If  the 
mean  value  is  zero,  then  the  standard  deviation  is  equal  to  the  rms  value 
and  Eq  (5)  reduces  to  Eq  (2). 

FATIGUE  MODEL  COMPARISON  WITH  EXPERIMENTAL 

RESULTS 


The  peak  probability  density  function  (PPDF)  is  needed  or  preferably 
the  time  history  from  strain  or  dynamic  response  measurements  to 
evaluate  the  fatigue  model  developed.  Also  needed  are  sinusoidal  8  -N 
curves  for  the  structure,  and  knowledge  of  its  boundary  conditions  and  the 
equivalent  multimodal  cyclic  frequency. 

Two  types  of  peak  probability  density  techniques  were  investigated  from 
multimodal  nonlinear  strain  responses  [3].  These  were  called  major  peaks 
and  minor  peaks.  The  major  peaks  were  counted  for  the  largest  peaks 
between  zero  crossings.  The  nunor  peaks  were  counted  for  all  stress 
reversals  or  a  positive  slope  in  the  time  history  followed  by  any  negative 
slope.  The  effective  cyclic  frequency  is  much  lower  for  the  major  peak  coimt 
than  the  nunor  peak  coimt  However,  the  peak  probability  density 
functions  or  PPDFs  compared  for  these  two  cases  were  almost  the  same. 
The  major  peak  method  was  selected  for  further  study. 

Comparison  with  Beam  Data: 

The  K  and  a  terms  were  calculated  from  8-N  and  S-N  data,  where  S 
is  stress,  using  Eq  (3).  Selecting  two  values  of  strain  and  their 
corresponding  cycles  to  failure,  yields  two  simultaneous  equations 
which  were  solved  for  K  and  a.  Two  sinusoidal  E-N  curves  for 
BS1470-NS3  aluminum  alloy  which  has  a  relatively  low  tensile  strength 
were  obtained  from  Bennouna  and  White  [4  Fig  8].  These  were  for  a 
cantilevered  beam  and  a  clamped-clamped  (C-C)  beam  as  shown  in  Fig 
1.  The  K  and  a  terms  calculated  were  used  to  compute  the  cycles  to 
failure,  Nc,  for  each  strain  level.  Table  I  shown  is  the  same  as  Table  I  in 
reference  [4]  except  was  calculated  from  Eq  (3)  to  sum  the  damages. 


971 


Delta  is  the  sample  size.  The  cyclic  frequency  was  for  one  mode  the  one 
resonant  response  frequency.  The  time  to  failure  in  hours  using  Eq  (2) 
for  the  cantilevered  beam  was  16.6  hours  compared  with  16.2  predicted 
theoretically  [4],  15.3  and  15.9  obtained  experimentally  [4].  Both 
theoretical  results  were  essentially  equal,  but  slightly  higher  than  the 
test  results.  The  time  to  failure  from  Eq  (2)  for  the  (C-C)  beam  was  3.04 
hours  compared  with  2.53  theoretically  [4]  and  5.25  and  5.92 
experimentdly  [4].  Both  theoretical  results  were  about  one  half  of  the 
test  results.  The  K  and  a  terms,  the  theoretical  fatigue  life  times  and  the 
experimental  fatigue  life  times  are  listed  in  Table  H.  The  table  contains 
three  sections:  lifetimes  calculated  using  a  strain  gauge  PPDF,  a 
displacement  PPDF  and  the  Gaussian  and  Rayleigh  PDFs.  As  noted  in 
reference  [4],  failure  occurred  much  earlier  for  the  C-C  beam  than  the 
cantilevered  beam  for  the  same  strain  level.  This  was  attributed  to  the 
influence  of  a  large  axial  strain  in  the  clamped-  clamped  beam. 

Comparison  with  Plate  Data: 

Two  fatigue  tests  were  conducted  to  provide  some  additional  limited 
data  for  comparison  with  the  fatigue  model  developed.  These  tests 
used  the  base  excitation  method  with  a  1.09x10®  N  (20,000  Ibf) 
electrod)mamic  shaker.  The  clamping  fixture  consisted  of  a  flat 
aluminum  alloy  6061-T6  plate  19  mm  thick  and  four  clamping  bars  of 
equal  thickness.  The  radius  of  curvature  of  the  clamping  edges  was  4.76 
mm  to  prevent  early  fatigue  failure.  A  four  bar  clamping  arrangement 
was  selected  to  prevent  buckling  of  the  plate  while  torquing  the 
clamping  bolts.  The  undamped  size  was  254  x  203  x  1.30  mm  which 
results  in  a  1.25  aspect  ratio.  Strain  gauges  were  bonded  along  the  center  of 
the  larger  dimension  (SG  2)  and  at  the  center  of  the  plate  (SG  3). 
Displacements  were  measured  with  a  scanning  laser  vibrometer  at  the 
center  of  the  plate.  An  accelerometer  was  moimted  on  the  shaker  head  to 
determine  the  acceleration  imparted  to  the  damped  plate.  A  flat 
acceleration  spectral  density  was  used  between  100-1500  Hz.  Recordings 
were  taken  at  increasing  levels  of  exdtation  up  to  the  fatigue  test  level.  The 
time  to  detecting  the  first  fatigue  crack  was  recorded  for  each  plate. 

The  constants  K  and  a  were  calculated  from  random  single  mode  S-N 
data  for  7075-T6  aluminum  alloy  [5  p  489]  shown  in  Fig  1,  with 

K  =  1.01x10^^  and  a  =  -  0.175.  The  rms  stress  ( Srms )  ^^s  changed  to  rms 
strain,  Srms  =  EBrms,  where  E  is  Young's  modulus.  The  stress  was 
measured  half  way  between  two  rivets  along  the  center  line  between  the 
rivets  on  the  test  specimen.  The  strain  gauge  location,  stress 
concentrations,  and  the  boundary  conditions  greatly  affect  the  strain 


972 


level  measured.  Correction  factors  are  needed  for  a  different  set  of 
conditions  and  to  convert  random  data  to  sinusoidal  data.  Sinusoidal 
£  -N  bending  coupon  curves  for  7075-T6  aluminum  alloy  were  difficult  to 
find.  S-N  curves  were  found  for  an  aerospace  material  with  both 
sinusoidal  and  random  excitations.  These  curves  were  nearly  parallel. 
The  sinusoidal  strain  was  1.38  times  larger  than  the  random  strain  for 

10^  cycles.  Multiplying  the  constant  K  for  the  7075-T6  material  by  1.38 
resulted  in  K  =  1.40x10^^ . 

Early  strain  gauge  failures  prevented  strain  measurement  above  500 
microstrain  with  20.7  g  rms  shaker  excitation.  The  fatigue  test  level  was 
115  g  rms  and  the  response  contained  at  least  six  frequency  response 
peaks.  The  major  peak  strain  PPDFs  were  determined  for  5.32  g  rms 
and  20.7  g  rms  as  shown  in  Fig  2  with  the  Gaussian  PDF.  Compared 
with  the  Gaussian  distribution,  an  increased  number  of  peaks  occurred 
greater  than  1  sigma  and  smaller  than  -1  sigma.  Also  a  larger  number  of 
peaks  occurred  around  zero.  The  PPDF  determined  from  the  20.7g  rms 
test  case  was  used  to  predict  fatigue  life,  but  a  new  strain  estimate  was 
needed  since  the  excitation  level  increased  5.6  times.  The  displacement 
is  directly  related  to  the  strain  at  each  excitation  level.  Displacement 
measurements  at  the  fatigue  test  level  were  used  to  estimate  the  strain 
level  shown  in  Fig  3.  The  estimated  strain  from  the  figure  was  770 
microstrain  for  SG  2.  The  scale  of  the  displacement  measurements  was 
adjusted  to  coincide  with  strain  measurements  at  increasing  increments 
of  shaker  excitation. 


The  equivalent  cyclic  multimodal  frequency  is  needed  to  predict  the 
fatigue  life.  Prediction  of  the  linear  modal  frequencies  is  carried  out  by 
a  variety  of  methods.  Usually  the  first  mode  prediction  is  the  most 
accurate.  The  cyclic  multimodal  nonlinear  frequencies  have  been 
studied  for  two  clamped  beams  and  two  clamped  plates  [3].  These  were 
based  upon  the  peak  probability  density  functions  (PPDFs)  where  the 
peaks  were  counted  for  a  specific  time  interval,  from  which  the 
nonlinear  cyclic  multimodal  frequencies  were  calculated.  Generally  the 
resonant  frequencies  increased  with  increasing  excitation  levels.  Those 
for  the  two  beams  increased  more  rapidly  than  those  for  the  two  plates. 
Very  little  change  was  noted  for  the  plates.  The  equivalent  cyclic 
miiltimodal  frequency  determined  via  Eq  (4)  from  the  SG  2  PPDF  was 
348  Hz. 

The  time  to  failure  in  hours  predicted  using  Eq  (5)  and  the  parameter 
mentioned  above  for  the  clamped  plate  was  0.706  hours  compared  with 
1.17  and  0.92  shown  in  Table  11.  The  predicted  result  was  slightly  lower 


973 


than  the  test  results.  A  Srms’^  curve  was  calculated  with  the  test 
lifetimes  available,  by  determining  a  new  constant  K,  assuming  the 
slope  was  the  same  as  for  the  riveted  coupon  and  applying  the 
sinusoidal  correction  factor.  The  time  to  failure  in  hours  predicted 
using  Eq  5  and  the  calculated  e-N  curve  for  the  clamped  plate  was 
0.274  hours  compared  with  1.17  and  0.92  shown  in  Table  E.  The  cyclic 
multimodal  frequency  used  was  the  same  at  that  determined  from  SG  2 
and  the  same  strain  was  used.  This  prediction  was  about  1/3  of  the  test 
results.  This  method  incorporates  the  failure  data  at  two  points. 

The  displacement  PPDF  shown  in  Fig  4  was  used  to  predict  fatigue 
life.  The  number  of  displacement  peaks  increased  significantly  above 
the  strain  PPDF  around  1  sigma  and  -1  sigma.  The  large  number  of 
peaks  around  zero  was  similar  to  the  characteristics  observed  in  the 
strain  PPDF.  The  equivalent  cyclic  multimodal  frequency  was  375  Hz, 
slightly  higher  than  that  determined  from  the  strain  PPDF.  However, 
the  same  frequency  (348  Hz)  was  used  to  predict  fatigue  life.  The  time  to 
failure  in  hours  using  Eq  (5)  for  the  clamped  plate  was  1.15  hours  with 
the  riveted  e-N  curve  and  0.446  with  the  calculated  8-N  curve  shown 
in  Table  E.  This  PPDF  improved  considerably  the  prediction.  The 
Gaussian  and  Rayleigh  PDFs  were  used  to  predict  the  time  to  failure 
with  the  same  parameters  as  those  used  with  the  riveted  e-N  curve. 
The  lifetime  using  the  displacement  PPDF  was  1.15  hours,  using  the 
Gaussian  PDF,  0.600  hours  and  using  the  Rayleigh  PDF,  0.237  hours,  as 
shown  in  Table  E.  The  Gaussian  PDF  under  predicts  by  a  factor  of  2. 
The  Rayleigh  PDF  under  predicts  by  a  factor  of  5. 

The  spread  sheets  containing  PPDF  /  Ncm  data  for  various  sigma 
values  were  used  to  determine  damage  accumulation  shown  in  Fig  5. 
Almost  55%  of  the  normalized  damage  occurs  between  -2  and  -1  sigma 
and  38%  of  the  damage  between  1  and  2  sigma  using  the  displacement 
PPDF.  However,  the  damage  is  spread  more  evenly  using  the  available 
strain  gauge  PPDF.  The  strain  gauge  PPDF  was  recorded  at  a  much 
lower  level  than  the  displacement  PPDF.  The  damage  accumulation 
compared  more  closely  to  Gaussian  PDF  than  the  Rayleigh  function. 


Damage  Model  with  a  Specific  Function  Describing  the  PPDF: 

A  curve-fitting  routine  was  used  to  determine  a  mathematical  function 
for  a  high  level  strain  gauge  PPDF  for  the  clamped  shaker  plate.  The 
most  important  part  of  the  fit  is  outside  the  range  of  -1  to  1  sigma,  since 
most  of  the  damage  accumulation  occurs  outside  this  range.  The 
highest  ranking  function  was  a  tenth  order  polynomial  followed  by 


974 


ninth  and  eighth  order  polynomial  fits.  The  goodness  of  fit  in  order 
from  1  to  14  ranges  from  0.9775  to  0.9625,  which  are  very  close 
statistically.  The  tenth  order  polynonual  is, 

y  =  a  4*  bx  +  cx^  +  dx^  +  ex"^  +  fx^  +  gx^  +  hx^  +  ix^  +  jx^  +  kx^^  (6) 

where  a=0.346,  b=-0.0148,  c=-0.137,  d=-0.054,  e=0.090,  f=0.043,  g=- 
0.0400,  h=-0.00976,  i=  0.00722,  j=6.85xl0"‘,  k=<4.36xl0'"  .  Ranked 
fourteenth  is  a  natural  logarithmic  function.  The  function  and  its 
coefficients  are, 

Iny  =  a  +  bx  +  cx^  +  dx^  +ex^  4-fx^  (7) 


where  a=-1.088,  b=-0.1191,  c=-0.1.302,  d=0.0104,  e=-0.0653,  and  f=0.0079. 
The  function  fits  the  test  data  similar  to  the  tenth  order  polynomial  and 
may  be  easier  to  use.  Ranked  forty-first  is  a  Gaussian  function.  The 
function  and  its  coefficients  are. 


y  =  a  +  b  exp|o.5[(x  -  c)  /  d]^  | 


(8) 


where  a=-0.0968,  b=0.4485,  c=-0.050  and  d=1.45.  The  function  fits  better 
for  sigma  values  of  2  or  greater  than  those  of  -2  sigma  and  greater.  A 
constant  coefficient  is  used  to  fit  the  Gaussian  function  to  permit  shifting 
the  function  to  fit  the  test  data.  This  equation  can  be  used  in  the  PPDF 
in  Eq  (5), 

a  +  b  exp  [-0.5 [(x  -  c)  /  d]^  } 

-  [K/(e)f“ 


3600£ 


■cm 


(9) 


where  a=-0.0968,  b=0.4485,  c=-0.050  and  d=1.45. 


CONCLUSIONS 

The  prediction  of  multimodal  fatigue  life  is  primarily  dependent  upon 
the  peak  probability  density  function  (PPDF)  which  changes  shape  with 
increasing  excitation  levels.  The  next  in  order  of  importance  is  the 
sinusoidal  e  -N  bending  fatigue  curve  and  finally  the  effective  multimodal 
cydic  frequency. 

A  multimodal  fatigue  model  was  developed  with  the  PPDF  estimated 
from  a  form  of  the  Gaussian  function  being  useful  especially  in  the 


975 


range  of  cydes  to  failure  from  10^-10^.  The  lifetime  predication 
calailations  for  the  damped-clamed  beam  was  about  one  half  the 
experimental  value.  For  the  plate,  the  calculations  was  about  one  half 
the  experimental  value.  Using  riveted  coupon  fatigue  data,  the 
calculation  was  accurate. 


REFERENCES 

1.  B.L.  Clarkson,  April  1994,  '‘Review  of  sonic  fatigue  technolog/', 
NASA  contractor  report  4587,  NASA  Langley  Research  Center, 
Hampton,  Virginia. 

2.  R.G.  White,  October  1978,  "A  comparison  of  some  statistical 
properties  of  the  responses  of  aluminium  alloy  and  CFRP  plates  to 
acoustic  excitation".  Composites  9(4),  125-258. 

3.  H.F.  Wolfe,  October  1995,  "An  experimental  investigation  of 
nonlinear  behaviour  of  beams  and  plates  excited  to  high  levels  of 
dynamic  response",  PhD  Thesis,  University  of  Southampton. 

4.  M.  M.  Bennouna,  and  R.  G.  White,  1984,  "The  effects  of  large 
vibration  amplitudes  on  the  dynamic  strain  response  of  a  clamped- 
clamped  beam  with  consideration  on  fatigue  life".  Journal  of  Sound  and 
Vibration,  96  (3),  281-308. 

5.  J.  R.  Ballentine,  F.  F.  Rudder,  J.  T.  Mathis  and  H.E.  Plumblee, 

1968,  'Refinement  of  sonic  fatigue  structural  design  criteria",  AFFDL  TR 
67-156,  AD831118,  Wright-Patterson  AFB,  Ohio. 


TABLE  I 

FATIGUE  CALCULATIONS  USING  EQUATION  5.9 

CANTILEVERED  BEAM  BS  1470-NS3  (REF  4  FIG  8) 


e/sd 

sd 

P£ 

e 

ll£ 

Nc=(2.172 

xl0“/sd)“““ 

PPDF  X  A 

PPDF  X  A/Nc 

t 

hours 

0.5 

H 

213 

15079045.41 

n 

1.41256E-07 

1 

|g 

425 

4390936.857 

mam 

7.06000E-07 

1.5 

425 

638 

2125446.064 

WBrn 

1.12917E-06 

2 

425 

850 

1273261.536 

0.138 

1.08383E-06 

2.5 

425 

1063 

854026.6252 

0.060 

7.02554E-07 

3 

425 

1275 

617189.6449 

0.018 

2.91645E-07 

3.5 

TOTAL 

425 

1488 

468372.0662 

0.005 

0.984 

1.06753E-07 

4.16121E-06 

16.7 

976 


TABLE  n 

SUMMARY  OF  FATIGUE  CALCULATIONS 


Figure  1  8  -N  curves  for  aluminum  alloys  tested. 


Figure  2  Normalized  strain  PPDF  comparison  with  a  Gaussian  PDF. 


978 


NORMALIZED  DISPLACEMENT 
MOJORPPDF 


2 


Figure  4  Normalized  displacement  PPDF  comparisons, 


9 


980 


ACOUSTIC  FATIGUE  AND  DAMPING  TECHNOLOGY  IN 
COMPOSITE  MATERIALS 


By  B.  Benchekchou  and  R.G.  White 


Abstract 

Considerable  interest  is  being  shown  in  the  use  of  composite  materials  in 
aerospace  structures.  Important  areas  include  development  of  a  stiff,  lightweight 
composite  material  with  a  highly  damped,  high  temperature  polymer  matrix 
material.  The  study  described  in  this  paper  concerns  the  application  of  such 
material  in  the  form  used  in  thin  skin  panels  of  aircraft  and  investigation  of  its 
fatigue  properties  at  room  and  high  temperature.  For  this  purpose,  flexural  fatigue 
tests  have  been  carried  out  at  two  different  temperatures  and  harmonic 
three-dimensional  FE  analyses  were  performed  in  order  to  understand  the 
dynamic  behaviour  of  plates.  Random  acoustic  excitation  tests  using  a 
progressive  wave  tube,  up  to  an  overall  sound  pressure  level  of  162  dB,  at  room 
temperature  and  high  temperatures  were  also  performed  in  order  to  investigate  the 
dynamic  behaviour  of  panels  made  of  the  materials.  Various  methods  for 
including  damping  in  the  structure  were  examined  when  parameter  studies  were 
carried  out,  and  conclusions  have  been  drawn  concerning  optimal  incorporation 
of  a  highly  damped  matrix  material  into  a  high  performance  structure. 

1-Introduction 

Significant  areas  of  primary  and  secondary  structures  in  military  aircraft  operate 
at  high  temperature  and  are  subjected  to  high  levels  of  random  acoustic  loading, 
because  of  their  closeness  to  jet  effluxes.  There  is  then  a  need  to  develop  a  carbon 
fibre  reinforced  plastic  material  with  a  high  temperature  polymer  matrix  and  high 
fatigue  resistance.  Highly  damped  composite  structures  should  be  developed  in 
order  to  better  resist  dynamic  loading  and  to  have  an  enhanced  fatigue  life.  Work 
previously  carried  out  on  improving  the  damping  in  fibre  reinforced  plastic  (FRP) 
composites  as  well  as  the  number  of  approaches  which  can  be  taken  to  improve 
the  damping  properties  of  polymeric  composites  have  been  summarised  in  [1]. 
The  aim  of  the  research  described  here  was  to  study  lightweight  composite 
materials  with  a  highly  damped,  high  temperature  polymer  matrix  material,  by 


981 


investigating  its  mechanical  and  acoustic  fatigue  properties,  the  latter  investigation 
being  carried  out  using  thin,  multilayered  plates. 

2-ExperimentaI  work 

For  this  type  of  study,  two  adequate  prepregs  were  highlighted  after  investigation: 
SE300  and  PMR15.  The  SE300  material  was  carbon  fibre  reinforced  prepreg  of 
(0°/90°)  woven  form,  0.25  mm  thick  and  had  60%  fibre  volume  fraction,  with  no 
suitable  data  available  on  the  material  properties.  Dynamic  mechanical  thermal 
analyser  (DMTA)  analyses  carried  out  on  four  specimens  (  20  nun  long  and  12 
mm  wide)  with  different  lay-ups  i.e.  (0°/90°)4,  (+45°/-45°)4,  (0°/90°,  45°/45°)s 
and  (45®/45°;0®/90°)s,  allowed  to  get  provided  the  material  properties.  Results 
from  DMTA  analyses  are  shown  in  Table  l.a  where  the  loss  factor  and  the 
Young’s  modulus  values  at  40^C  and  at  the  glass  transition  temperature  Tg  are 
presented.  The  loss  factor  values  varied  from  0.0097  to  0.085  for  a  range  of 

temperature  from  40®C  to  300®C. 

The  PMR15  prepreg  was  also  of  (0^/90^)  woven  carbon  form  and  had  58% 
volume  fraction.  Six  DMTA  specimens  having  the  following  lay-ups:  (0®/90®)4, 
(00/90^)8,  (4450/-450)4,  (+450/-45<^)8,  (0^/90^-,  450/450)s  and 

(450/450.00/900)^  were  made.  Results  from  DMTA  analyses  showed  that  the  loss 
factor  values  varied  from  0.0129  to  0.0857  for  a  range  of  temperature  from  40®C 
to  400®C,  with  a  value  of  loss  factor  of  0.1293  at  375®C,  the  maximum 
temperature  for  normal  use  being  352^C.  The  loss  factor  and  Young’s  modulus 
values  at  40®C  and  Tg  are  given  in  Table  l.b. 

Mechanical  behaviour  of  the  selected  materials 

The  fatigue  characteristics  of  these  new  materials  were  investigated  and  results 
were  compared  with  those  of  well  established  structural  materials.  Mechanical 
fatigue  tests  of  SE300  and  PMR15  samples  using  "sinusoidal"  loading  at  a  chosen 
maximum  strain  level,  i.e.  8000  pS  were  carried  out  and  performances  compared 
to  that  of  an  XAS/914  sample.  A  mechanical  (flexural)  fatigue  rig  was  used  for 
this  purpose  to  test  specimens  in  a  cantilevered  configuration.  Details  of  the  rig  are 
available  in  [2].  The  particular  clamp  used  was  designed  by  Drew  [3]  to  induce 
damage  in  the  centre  of  the  specimen  instead  of  having  edge  damage,  i.e.  peeling 
while  flexural  tests  are  carried  out.  In  order  to  investigate  the  performance  of  these 
new  materials  at  high  temperature,  fatigue  tests  were  also  carried  out  on  samples, 

at  210^C  .  This  was  achieved  by  using  a  heating  system  which  consisted  of  two  air 


982 


blowers  (electronically  regulated  hot-air  guns)  positioned  at  40  mm  above  and 
below  the  specimens,  which  allowed  specimens  to  be  tested  at  a  uniform  temperature 

of  210±5°C.  The  aim  of  the  mechanical  fatigue  test  was  to  determine  the  number  of 
loading  cycles  needed  for  damage  to  occur  and  its  subsequent  growth  rate  in 
cyclically  loaded  composite  specimens  of  SE300  and  PMR15  matrix  materials  .  The 
samples  were  140  mm  long,  70  mm  wide  and  2  mm  thick.  Fatigue  tests  of  SE300  (S3 
and  S4)  and  PMR15  (PI  and  P2)  specimens,  at  room  temperature  and  at  210OC 
respectively,  at  a  level  of  8000  p.S,  located  by  the  peak  of  the  half-sine  clamp,  have 
been  carried  out.  Ultrasonic  scans  of  specimens  S3  and  S4  before  any  loading  cycles 
and  after  100,  500,  1000,  2000,  5000,  10000,  20000  and  50000  loading  cycles  are 
shown  respectively  in  Figures  l.a-h  and  Figures  2.a-h.  A  small  delamination, 
indicated  by  lighter  areas  in  the  scans,  starts  to  show  in  both  specimens  S3  and  S4 
after  applying  500  loading  cycles  and  increases  substantially  after  5000  loading 
cycles.  After  5000  loading  cycles,  the  damage  area  increased  more  for  specimen  S4 
than  specimen  S3,  which  shows  that  the  latter  is  slightly  more  fatigue  resistant.  In 
other  words,  when  increasing  the  temperature  from  250C  to  210<^C,  the  resistance  to 
fatigue  slightly  decreases.  Figures  3.a-h  and  Figures  4.a-h  show  the  ultrasonic  scans 
for  PMR15  specimens  PI  and  P2  before  and  after  several  loading  cycles.  For  both 
specimens  PI  and  P2,  damage  starts  after  500  loading  cycles  and  increases 
substantially  after  20000  loading  cycles.  At  this  stage,  delamination  areas  are  similar 
for  both  specimens  PI  and  P2  and  just  a  little  more  pronounced  in  specimen  P2, 
which  shows  that  the  latter  is  slightly  less  fatigue  resistant.  Hence,  an  increase  in 
temperature  leads  to  a  decrease  in  the  fatigue  resistance  properties  of  the  specimens. 
From  Figures  1,  2,  3  and  4,  one  can  conclude  that  the  PMR15  specimens  tested  were 
slightly  more  fatigue  resistant  than  the  SE300  specimens.  In  fact,  damage  in 
specimens  was  generally  more  defined,  clearer  and  spread  more  rapidly  in  the  SE300 
samples  than  was  the  case  for  the  PMR15  samples.  Figure  5  shows  damage 
propagation  occuring  in  an  XAS/914  sample  (XI),  with  (0o/±45®/0O)s  stacking 
sequence,  tested  at  8000  |iS  level  and  at  room  temperature,  from  [3].  Note  that 
substantial  damage  existed  after  1000  loading  cycles  in  this  specimen,  which  shows 
that  both  SE300  and  PMR15  are  more  fatigue  resistant  than  XAS/914  at  room 
temperature. 

Acoustic  fatigue  behaviour  of  panels  of  the  selected  materials 
Investigations  were  carried  out  by  installing  the  CFRP  plates  in  an  acoustic 
progressive  wave  tunnel,  (APWT)  in  order  to  determine  the  response  of  CFRP  plates 
under  broadband  acoustic  excitation  simulating  jet  noise.  The  plate  was  fully 
clamped  around  its  boundaries  on  to  a  vertical  steel  frame  fixed  to  one  side  of  the 
APWT,  so  that  it  formed  one  of  the  vertical  walls  of  the  test  section  of  the  APWT. 


983 


Overall  sound  pressure  levels  (OSPL)  up  to  165  dB  of  broadband  noise  in  the  test 
section  of  the  tunnel  was  generated  by  a  Wyle  Laboratories  WAS  3000  siren.  A 
heater  panel  capable  of  heating  and  maintaining  the  temperature  of  test  plates  up  to 

300*^0  while  mounted  on  the  tunnel  was  designed  and  built.  Temperatures  were 
monitored  and  controlled  via  thermocouples  on  the  panel.  Plates  were  excited  by 
broadband  excitation  in  the  frequency  range  80-800  Hz.  A  B&K  type  4136 
microphone  mounted  at  the  centre  of  the  test  section  of  the  tunnel  adjacent  to  the 
mid-point  of  the  plate  were  used  for  sound  pressure  measurements.  Eight  strain 
gauges,  four  on  each  side  of  the  plate  were  attached  in  order  to  monitor  the  strain 
distribution  in  the  panel  while  the  rig  was  running;  more  details  of  the  experiment 
may  be  found  in  [1].  Acoustic  tests  were  run,  at  various  temperatures  and  OSPL. 
Since  plates  were  excited  in  the  frequency  range  80-800  Hz,  spectral  analyses 
would  not  include  the  first  natural  frequency.  The  natural  fundamental  frequency  of 
an  SE300  clamped  panel  was  found  to  be  49.02  Hz  analytically.  The  second  -and 
third  resonance  frequencies  were  149.5  and  198.5  Hz,  as  calculated  from  strain 
spectral  densities,  from  tests  carried  out  with  an  OSPL  of  156  dB  and  a  temperature 
of  1620C.  At  162  dB,  results  showed  that  the  second  resonance  frequency  was  113 
Hz  at  150OC  and  106.5  Hz  at  195^0,  which  shows  that  when  the  temperature  and 
the  OSPL  increased,  the  resonance  frequencies  of  the  plates  decreased.  Also,  it  was 
found  that  the  damping  increased  at  elevated  temperatures.  The  overall  modal 
viscous  damping  ratios,  for  the  second  mode,  were  calculated  from  strain  spectral 
densities,  for  an  SE300  panel  driven  at  an  OSPL  of  162  dB  and  at  1950C,  and  was 
found  to  be  8.91%;  this  value  is  similar  to  that  calculated  from  analytical 
simulations,  for  the  first  mode,  which  is  8,50%  at  242^0,  (see  the  analytical  section 
below).  A  typical  strain  spectral  density  obtained  from  recorded  results  is  presented 
in  Figure  6  for  an  SE300  specimen,  at  an  OSPL  =  156  dB  and  at  1620C,  from  a 
strain  gauge  in  the  centre  of  the  specimen.  Maximum  RMS  strain  values  recorded 
from  experimental  tests,  at  a  strain  gauge  in  the  centre  of  the  specimen  were,  at  an 
OPSL  =156  dB,  1300  \iS,  1800  \iS  and  2800  pS  at  room  temperature  and  at  90^0 
and  1620C  respectively.  These  results  clearly  indicate  a  trend  for  significant 
increase  in  dynamic  response  with  increasing  temperature. 

Experiments  were  also  carried  out  on  a  PMR15  panel  at  various  OSPL  and 
temperatures.  Results  from  tests  run  at  159  dB  and  at  room  temperature  show  that 
the  second  and  third  resonance  frequencies  were  indicated  as  112  and  182.5  Hz. 
When  the  OSPL  increased  to  162  dB,  the  second  and  third  resonance  frequencies 
decreased  to  110.5  Hz  and  176.5  Hz.  At  the  same  OSPL  (162dB)  and  when 
temperature  increased  to  2810C,  the  third  resonance  frequency  became  139  Hz.  This 


984 


shows  that,  for  PMR15  plates,  when  the  temperature  and  the  OSPL  increased,  the 
resonance  frequencies  of  the  plates  decreased.  It  was  also  clear  that  modal  damping 
increased  with  increasing  temperature.  In  fact,  the  overall  viscous  damping  reached 
20%,  for  the  second  mode,  at  an  OSPL=162  dB  and  at  2810C.  It  must  be  stated  here, 
however,  that  apparent  damping  trends  could  include  nonlinear  effects  which 
influence  bandwidths  of  resonances.  Maximum  RMS  strain  values  recorded  at  room 
temperature,  by  a  strain  gauge  in  the  centre  of  the  plate  were  found  to  be:  2700  pS  at 
153  dB,  2800  pS  at  157.9  dB  and  2900  pS  at  159  dB. 

It  is  clear  from  these  values  that  increasing  the  OSPL  obviously  leads  to  an  increase 
of  the  strain  in  the  plate.  Similar  results  were  observed  when  the  temperature  was 
increased.  In  fact,  at  an  OSPL  of  162  dB,  the  maximum  strain  values  recorded  by  a 
strain  gauge  in  the  centre  of  the  specimen  were  3000  pS,  3400  pS  and  5000  pS  at 
105^0,  1650C  and  281^0  respectively,  which  clearly  indicates  the  effects  of 
temperature.  It  was  observed  that  both  the  PMR15  and  SE300  panels  behaved  in  a 
non  linear  manner. 

Attempts  to  acoustically  fatigue  a  PMR15  panel  were  made  at  162  dB.  No  signs  of 
fatigue  damage  were  shown  in  an  ultrasonic  scan  of  the  panel  after  1389  minutes  of 
running  time. 


3-Analytical  work 

In  order  to  examine  various  methods  for  including  damping  in  a  structure,  parametric 
studies  were  carried  out  using  the  finite  element  FE  method.  ANSYS  software  has 
been  used.  A  three-dimensional,  3D  layered  element,  SOLID46  was  used  to  build 
theoretical  models.  The  element  is  defined  by  eight  nodal  points,  average  layer 
thickness,  layer  material  direction  angles  and  orthotropic  material  properties,  [4]. 
Meshes  were  built  in  order  to  carry  out  modal  and  harmonic  analyses  of  multilayered 
composite  plates  (410  mm,  280  mm,  2  mm).  The  plates  were  fully  clamped  along  all 
edges,  in  order  to  simulate  the  panels  tested  in  the  APWT.  Natural  frequencies  were 
first  determined  from  free  vibration  analyses  and  compared  to  resonance  frequency 
values  derived  from  experimental  data.  Then,  the  plate  was  driven  by  harmonic 
loading  at  one  point  of  application.  The  forcing  frequency  varied  from  0  to  400  Hz. 
The  amplitude  of  the  load  was  50  N.  Results  for  displacements  and  response  phase 
angles  relative  to  the  force  for  a  chosen  position  on  the  plate  as  a  function  of 
frequency  were  obtained.  The  approach  was  then  to  carry  out  parameter  studies  in 
order  to  examine  various  methods  for  including  damping  in  the  structure,  i.e.  to  use 
highly  damped  matrix  material  throughout  the  whole  structure  or  possible 


985 


incorporation  in  a  few  layers.  Structural  damping  was  included,  allowing  models  to 
run  with  different  damping  values  in  each  ply  of  the  panel.  Structural  damping  is 
inherent  in  the  structure  and  depends  on  the  natural  frequency;  details  on  structural 
damping  modeling  may  be  found  in  [1].  Analyses  were  performed  considering 
structural  damping  for  the  first  mode.  The  structural  damping  was  then  varied  for 
plies  with  the  same  orientation  for  a  viscous  damping  ratio  ^  =  0.01, 0.02,  0.05,  0.10 
and  0.20. 

Simulations  with  SE300 

Models  were  built  up  with  the  following  stacking  sequence  ((45°/45°),(0°/90°))s, 
lay-up  used  for  the  experimental  plates.  Table  2  gives  the  first  three  modal 
frequencies  of  the  panel  obtained  from  free  vibration  analyses  results.  Harmonic 
simulations  were  carried  out  and  the  overall  damping  value  was  calculated  for  each 
case  with  results  given  in  Table  3.  As  can  be  seen,  if  high  overall  damping  is  needed 
for  a  structure  composed  of  the  SE300  material,  increasing  the  damping  value  of  the 
(45°/45®)  orientation  plies  most  significantly  increases  the  overall  damping  value  of 
the  panel.  In  fact,  putting  a  damping  value  of  20%  in  the  (45°/45°)  orientation  plies 
leads  to  an  overall  viscous  damping  value  of  14.52%,  which  is  better  than  including 
a  10%  damping  value  in  all  of  the  plies  of  the  structure. 

Harmonic  analyses  of  fully  clamped  plates  were  also  performed  with  the  values  of 
material  properties  taken  at  several  temperatures.  Simulations  were  carried  out  with 
material  properties  at  2420C  and  300OC.  Free  vibration  analyses  permitted 
calculation  of  the  modal  frequencies  of  the  panels  at  the  temperatures  mentioned 
above.  Table  2  also  lists  the  first  three  modal  frequencies  from  analyses  with 
material  properties  at  242^0  and  300^0.  The  overall  viscous  damping  values, 
obtained  from  FE  simulations,  are  given  for  each  ternperamre  in  Table  4.  Again,  the 
damping  value  has  been  varied  through  the  layers  and  the  overall  damping  value  was 
calculated  in  order  to  see  which  of  the  plies  contributes  the  most  to  heavily  damp  the 
plate.  It  was  found  that  putting  a  damping  value  of  20%  in  the  (45°/45°)  orientation 
plies,  the  first  mode  viscous  damping  ratios  were  14.62%  and  14.55%  at  242^0  and 
300OC  respectively.  This  shows  that  this  material  is  more  highly  damped  at  high 
temperature  and  presents  better  damping  properties  of  the  two  materials  at  242^0. 

Simulations  with  PMR15 

Free  vibration  analyses  of  models  built  up  with  the  following  stacking  sequence 
((45°/45°),(0°/90®))s  were  carried  out  and  the  first  three  modal  frequencies  of  the 
panel  are  shown  in  Table  5.  Harmonic  analyses  were  run  and  the  overall  damping 


986 


value  was  calculated  for  each  simulation  with  results  given  in  Table  6.  If  high 
overall  damping  is  needed  for  a  structure  composed  of  the  PMR15  material, 
increasing  the  damping  value  of  the  (45°/45®)  orientation  plies  most  significantly 
increases  the  overall  damping  value  of  the  panel.  In  fact,  putting  a  damping  value 
of  20%  in  the  (45°/45°)  orientation  plies  leads  to  an  overall  viscous  damping  value 
of  14.39%,  while  if  (0°/90®)  orientation  plies  have  a  20%  damping  value,  the 
overall  damping  is  7.42%. 

Harmonic  analyses  of  fully  clamped  plates  were  also  carried  out  with  the  values  of 
material  properties  taken  at  several  temperatures.  Simulations  were  carried  out 
using  material  properties  at  3750C  and  400oC.  Free  vibration  analyses  permitted 
calculation  of  the  modal  frequencies  of  the  panels  at  the  temperatures  mentioned 
above.  Table  5  lists  the  first  three  modal  frequencies  from  analyses  with  material 
properties  at  3750C  and  400^0.  The  overall  viscous  damping  values,  obtained 
from  FE  simulations,  are  given  for  each  temperature  in  Table  7.  Again,  the 
damping  value  has  been  varied  through  the  layers  and  the  overall  damping  value 
was  calculated  in  order  to  see  which  of  the  plies  contributes  the  most  to  heavily 

damp  the  plate.  It  was  found  that  putting  a  damping  value  of  20%  in  the  (45‘^/45°) 
orientation  plies,  the  first  mode  viscous  damping  ratios  were  18.39%  and  16.94% 
at  3750c  and  400OC  respectively.  This  shows  that  this  material  is  more  highly 
damped  at  high  temperature  and  presents  better  damping  properties  of  the  two 

materials  at  3750C. 


4~Conclusions 

Two  matrix  materials,  SE300  and  PMR15,  with  potential  for  use  in  aircraft 

structures  in  a  severe  environment,  i.e.  temperatures  up  to  SOO^C  were  selected  for 
this  study.  Material  properties  were  determined  using  DMTA  techniques  and 
results  show  that  these  materials  have  high  damping  abilities  at  high  temperature. 
Dynamic  loading  tests,  performed  in  flexure  at  room  and  high  temperature  showed 
that  the  carbon  fibre  reinforced  PMR15  material  is  more  fatigue  resistant  than 
SE300  and  XAS/914  based  composites.  Acoustic  tests  using  a  progressive  wave 
tunnel,  up  to  a  random  acoustic  OSPL  of  162  dB,  at  room  temperature  and 
elevated  temperamres  up  to  2810C  were  also  performed.  When  increasing  the 
excitation  level  and  the  temperature  higher  strain  values  in  the  centre  of  the  panels 
were  recorded.  Free  vibration  and  harmonic  FE  analyses  permitted  determination 
of  the  natural  frequencies  and  the  overall  viscous  damping  values.  Resonance 
frequencies  determined  from  results  obtained  from  acoustic  tests  were  similar  to 


987 


natural  frequencies  obtained  from  FE  simulations.  Overall  viscous  damping  values 
obtained  from  experimental  results  agreed  well  with  those  obtained  from  the  FE 
analyses  for  SE300  panels.  Results  obtained  for  PMR15  panels,  from  tests,  were 
higher  than  those  calculated  analytically.  Both  tests  and  simulations  showed  that 
SE300  and  PMR15  present  higher  damping  capabilies  at  high  temperatures. 
Conclusions,  via  parameter  studies  including  material  damping,  have  been  drawn 
concerning  optimal  incorporation  of  a  highly  damped  matrix  material  into  a  high 
performance  structure. 


5-AcknowIegments 

The  authors  wish  to  thank  the  Minister  of  Defence  for  sponsorship  of  the  programme 
of  research  under  which  the  work  was  carried  out.  Thanks  are  also  due  to  Dr  M.  Nash 
of  the  DRA,  Famborough  for  many  helpful  discussions  throughout  the  project.  _ 

6-References 

1-  Benchekchou,  B.  and  White,  R.G.,  Acoustic  fatigue  and  damping  technology  in 
FRP  composites,  submitted  to  Composite  Structures. 

2-  Benchekchou,  B.  and  White,  R.G.,  Stresses  around  fasteners  in  composite 
structures  in  flexure  and  effects  on  fatigue  damage  initiation:  I-Cheese-head  bolts. 
Composite  structures,  33(2),  pp.  95-108,  November  1995. 

3-  Drew,  R.C.  and  White,  R.G.,  An  experimental  investigation  into  damage 
propagation  and  its  effects  upon  dynamic  properties  in  CFRP  composite  material  . 
Proceedings  of  the  Fourth  International  Conference  on  Composite  Structures,  Paisley 
College  of  Technology,  July  1987. 

4-  ANSYS  theoretical  manual,  Swanson  Analysis  Systems  Inc,  December  1992. 


988 


Table  La:  Loss  factor  and  Young’s  modulus  values  at  40^C  and  at  Tg  for 
SE300  samples  analysed  by  the  DMTA. 


stacking  sequences 

(4.45/-45)4 

(-457+45)4 

(45/45;0/90)s 

(0/90)4 

Tg(°C) 

242 

242 

240.3 

238.71 

T|  atTg 

0.085 

0.085 

0.081 

0.061 

Ti  at  40°C 

0.012 

0.014 

0.010 

0.0097 

Log  E’  at40°C 

9.870 

9.840 

9.970 

10:097 

Table  Lb:  Loss  factor  and  Young’s  modulus  values  at  40^C  and  at  Tg  for 
PMR15  samples  analysed  by  the  DMTA. 


Stacking  sequences 

(+45/--45)4 

(0/90;45/45)s 

(45/45;0/90)s 

(0/90)4 

Tg  (OC) 

372 

375 

375 

375 

Tl  atTg 

0.117 

0.124 

0.121 

0.129 

T|  at  40®C 

0.0110 

0.0138 

0.0086 

0.0117 

Log  E’  at  40OC 

9.583 

9.944 

9.972 

9.875 

989 


Table  2:  The  first  three  modal  frequencies  for  SE300  panel;  analyses  carried  out 
with  material  propert  ies  at  room  temperature,  at  242*^0  and  at  300°C. 


Room  temperature 

242^0 

300OC 

49.02Hz 

44.72Hz 

43.61Hz 

155.80Hz 

143.27Hz 

140.17Hz 

212.04Hz 

194.19Hz 

189.68Hz 

Table  3:  Overall  viscous  damping  values  of  SE300  panel.  Values  are  calculated  from 
results  obtained  from  harmonic  analyses;  the  material  damping  being  considered  for 
the  first  mode. 


Simulation  with 
damping  of 

(45°/45°)orientation 

plies 

(0®/90°)orientation 

plies 

5% 

3.81% 

2.51% 

10% 

7.06% 

3.92% 

20% 

14.52% 

11.57% 

Table  4:  Overall  viscous  damping  values  of  SE300  panel,  for  the  first  mode. 


Temperature 

(°C) 

Overall  viscous  damping 

25 

1.20% 

242 

8.50% 

300 

5.45% 

990 


Table  5:  The  first  three  modal  frequencies  for  PMR15  panel;  FE  analyses  carried  out 
with  material  properties  at  room  temperature,  375°C  and  400°C. 


Room  temperature 

3750c 

400OC 

43.43Hz 

32.68Hz 

25.54Hz 

130.35Hz 

99.65Hz 

78.83Hz 

183.96Hz 

139.04Hz 

109.11Hz 

Table  6;  Overall  viscous  damping  values  of  PMR15  panel.  Values  are  calculated 
from  results  obtained  from  harmonic  analyses;  the  material  damping  being 
considered  for  the  first  mode.. 


Simulation  with 
damping  of 

(45°/450)orientation 

plies 

(0o/90O)orientation 

plies 

5% 

3.76% 

2.44% 

10% 

7.26% 

4.10% 

20% 

14.39% 

7.42% 

Table  7:  Overall  viscous  damping  values  of  PMR15  panel,  for  the  first  mode.. 


Temperature 

Overall  viscous  damping 

(OQ 

25 

1.33% 

275 

13.24% 

400 

8.6% 

991 


a:  before  any  loading  cycles  b:  after  100  loading  cycles  c:  after  500  loading  cycles 


d:  after  1000  loading  cycles  e:  after  2000  loading  cycles  f:  after  5000  loading  cycles 


g:  after  1 0000  loading  cycles  h:  after  20000  loading  cycles 


Figure  1.  Ultrasonic  scans  of  specimen  S3  after  applying  different  numbers  of  loading  cycles. 
(SE  300  material,  ambient  temperature) 


a:  before  any  loading  cycles  b:  after  100  loading  cycles  c:  after  500  loading  cycles 


d:  after  1000  loading  cycles  e:  after  2000  loading  cycles  f:  after  5000  loading  cycles 


Fisure  2.  Ultrasonic  scans  of  specimen  S4  after  applying  different  numbers  of  loading  cycles. 

(SE300  material,  210OC) 


992 


a:  before  any  loading  cycles  b:  after  100  loading  cycles  c:  after  500  loading  cycles 


d:  after  2000  loading  cycles  e:  after  20000  loading  cycles  f:  after  50000  loading  cycles 


g:  after  100000  loading  cycles 


Figure  3.  Ultrasonic  scans  of  specimen  PI  after  applying  different  numbers  of  loading  cycles 
(PMR15  material,  ambient  temperature,  8000]LlS) 


a:  before  any  loading  cycles  b:  after  500  loading  cycles  c:  after  1000  loading  cycles 


g:  after  10000  loading  cycles 


h:  after  20000  loading  cycles 


Figure  4.  Ultrasonic  scans  of  specimen  P2  after  applying  different  numbers  of  loading  cycles. 

(PMR15  material,  2100C,  8000M.S) 


993 


a:  before  any  b:  after  100  c:  after  500  d:  after  1000 

loading  cycles  loading  cycles  loading  cycles  loading  cycles 


e:  after  2000  f:  after  5000  g:  after  10000  h:  after  20000 

loading  cycles  loading  cycles  loading  cycles  loading  cycles 

Figure  5.  Ultrasonic  scans  of  an  X AS/9 14  specimen  fatigued  at  a  level  of  8000  llS 
showing  the  damage  propagation;  the  lay-up  is  (0/±45/90)s,  [3], 


0  Lin  Hz  RCLD  1.6k 


Figure  6:  SE300  specimen  SI  strain  spectral  density,  recorded  from  strain  gauge 
ST2,  OSPL  =156  dB,  temperature  =  162^C. 


994 


THE  BEHAVIOUR  OF  LIGHT  WEIGHT  HONEYCOMB  SANDWICH 
PANELS  UNDER  ACOUSTIC  LOADING 


David  Millar 
Senior  Stress  Engineer 
Short  Bros.  PLC 
Airport  Road 
Belfast 

Northern  Ireland 


SUMMARY 

This  paper  discusses  the  results  of  a  progressive  wave  tube  test  on  a  carbon  composite 
honeycomb  sandwich  panel.  A  comparison  was  made  with  the  test  panel  failure  and  the 
failure  of  panels  of  similar  construction  used  in  the  intake  ducts  of  jet  engine  nacelles. 
The  measured  panel  response  is  compared  with  traditional  analytical  methods  and  finite 
element  techniques. 


Nomenclature 

=  Overall  rms  stress  (psi)  or  strain  (jxs). 

7t  =3.14159 

=  Fundamental  frequency  (Hz). 

5  =  Critical  damping  ratio  (*0.017). 

Lps(fn)  =  Spectrum  level  of  acoustic  pressure  (-  expressed  as  a  fluctuating  rms  pressure  in  psi  in  a 
1  Hz  band). 

jr  =  Joint  acceptance  function  (non  dimensional). 

=  Characteristic  modal  pressure  (psi) 
ph  =  Mass  per  unit  area  (Ib/in^) 

Sic  “  Modal  stress  (psi)  or  modal  strain  (|j£). 

Wjc  =  Modal  displacement  (in), 

a  =  Panel  length  (in), 

b  =  Panel  width  (in). 

x,y,z  =  Co-ordinate  axes. 


1.0  Introduction 

Honeycomb  sandwich  panels  have  been  used  for  some  time  in  the  aircraft  industry  as 
structural  members  which  offer  a  high  bending  stiffness  relative  to  their  weight.  In 
particular,  they  have  proved  very  attractive  in  the  construction  of  jet  engine  nacelle 
intake  ducts  where,  in  addition  to  their  load  carrying  ability,  they  have  been  used  for 
noise  attenuation. 


995 


2.0  Acoustic  Fatigue 

The  intake  duct  of  a  jet  engine  nacelle  can  experience  a  severe  acoustic  environment 
and  as  such  the  integrity  of  the  nacelle  must  be  assessed  with  regard  to  acoustic  fatigue 
[1],  Acoustic  fatigue  characterises  the  behaviour  of  structures  subject  to  acoustic 
loading,  in  which  the  fluctuating  sound  pressure  levels  can  lead  to  a  fatigue  failure  of 
the  structure.  The  traditional  approach  to  acoustic  fatigue  analysis  has  assumed 
fundamental  mode  response  and  given  that  aircraft  panels  will  in  general,  have 
fundamental  frequencies  of  the  order  of  several  hundred  hertz,  it  is  clear  that  the 
potential  to  accumulate  several  thousand  fatigue  cycles  per  flight  can  exist. 

Techniques  for  analysing  the  response  of  structures  to  acoustic  loads  were  developed 
originally  by  Miles  [2]  and  Powell  [3],  Other  significant  contributions  are  listed  in 
References  4-7.  Design  guides  such  as  AGARD  [8]  and  the  Engineering  Sciences 
Data  Unit  (ESDU)  series  of  data  sheets  on  vibration  and  acoustic  fatigue  [9],  have 
proved  useful  in  the  early  stages  of  design. 

Note  -  further  details  on  the  general  subject  of  acoustic  fatigue  can  be  found  in  Ref 
10,  while  a  more  detailed  review  of  the  subject  up  to  more  recent  times  is  presented  in 
Ref  11. 


3.0  In  Service  Failures 

In  recent  years  a  number  of  failures  have  been  experienced  involving  intake  barrel 
honeycomb  sandwich  panels.  Failures  have  been  experienced  with  panels  which  had 
both  aluminium  facing  and  backing  skins  and  carbon  composite  panels.  The  metal 
intake  liner  was  observed  to  have  skin  cracking  and  also  core  failure,  while  the 
composite  panel  was  only  observed  to  have  core  failure. 

With  regard  to  the  metal  panels,  flight  testing  was  carried  out  and  the  predominant 
response  frequency  was  observed  to  be  at  the  fan  blade  passing  frequency  -  much 
higher  than  the  fundamental  frequency  of  the  intake  barrel;  this  went  some  way  to 
explaining  why  the  traditional  approach  in  estimating  the  response  did  not  indicate  a 
cause  for  concern.  The  response  of  the  panel  was  also  very  narrow  band  -  almost  a 
pure  sinusoid  (again  differing  from  the  traditional  approach  of  broad  band/random  load 
and  response),  and  the  subsequent  analysis  of  the  results  was  based  on  a  mechanical 
fatigue  approach  [12].  Subsequent  fleet  inspections  revealed  that  core  failure  was 
observed  prior  to  skin  failure  and  it  was  assumed  that  the  skin  failure  was  in  fact 
caused  by  a  breakdown  in  the  sandwich  panel  construction.  The  core  was  replaced 
with  a  higher  density  variety,  with  higher  shear  strength  and  moduli.  This  modification 
has  been  in  service  for  several  years  with  no  reported  failures.  The  modification 
represented  only  a  moderate  weight  increase  of  the  panel,  without  recourse  to 
changing  skin  thickness,  which  would  have  proved  very  expensive  and  resulted  in  a 
substantial  weight  penalty. 

As  mentioned  above,  another  intake  duct,  of  carbon  composite  construction,  also 
began  to  suffer  from  core  failure.  The  panels  of  this  duct  had  a  carbon  backing  skin 
while  the  facing  skin  had  a  wire  mesh  bonded  to  an  open  weave  carbon  sheet.  The  only 
similarity  was  the  use  of  the  same  density  of  honeycomb  core  (although  of  different 
cell  size  and  depth).  For  other  reasons  this  core  had  been  replaced  by  a  heavier  variety, 
prior  to  the  discovery  of  the  core  failures  and  the  impact  of  the  failures  was  minimised. 


996 


Limited  data  is  available  on  similar  failures  and  only  2  other  cases,  regarding  nacelle 
intake  barrels,  appear  to  have  been  documented  [13  &  14],  however  neither  case 
involved  sandwich  panels. 

A  number  of  theories  had  been  put  forward  as  to  the  cause  of  the  failures.  These 
included  neighbouring  cells  resonating  out  of  phase,  cell  walls  resonating  or  possibly 
the  panel  vibrating  as  a  2  degree  of  freedom  system  (the  facing  and  backing  skins 
acting  as  the  masses,  with  the  core  as  the  spring)  -  this  phenomenon  had  originally 
been  investigated  by  Mead  [15]. 


4.0  Physical  Testing 

A  number  of  tests  were  carried  out  with  "beam"  type  high  cycle  fatigue  specimens  and 
also  small  segments  of  intake  barrel.  None  of  these  tests  were  able  to  reproduce  the 
failures  observed  in  service  (Figure  1.0)  which  further  served  to  reinforce  the  belief 
that  the  failures  were  attributed  to  an  acoustic  mechanism  as  opposed  to  a  mechanical 
vibration  mechanism,  however  in  an  attempt  to  cover  all  aspects  it  was  decided  to 
carry  out  a  progressive  wave  tube  (PWT)  test  on  an  abbreviated  panel. 

For  simplicity  it  was  decided  to  test  a  flat  sandwich  panel  of  overall  dimensions 
36"x21"  (Figure  2.0).  The  panel  was  instrumented  with  12  strain  gauges  and  2 
accelerometers.  Two  pressure  transducers  were  also  mounted  in  the  fixture 
surrounding  the  specimen. 

Testing  was  carried  out  by  the  Consultancy  Service  at  the  Institute  of  Sound  and 
Vibration  Research  (IS  VR)  at  the  University  of  Southampton. 


4.1  PWT  Results 

The  panel  was  first  subject  to  a  sine  sweep  from  50  to  1000  Hz  in  order  to  identify  its 
resonant  frequencies.  The  response  of  a  strain  gauge  at  the  centre  of  the  panel  has  been 
included  in  Figure  3.0.  On  completion  of  the  sine  sweeps,  the  linearity  tests  were 
carried  out..  As  only  8  channels  could  be  accommodated  at  one  time,  it  had  been 
decided  to  arrange  the  parameters  into  5  groups,  with  each  group  containing  4  strain 
gauges,  2  accelerometers,  1  pressure  transducer  on  the  fixture  and  1  pressure 
transducer  in  the  PWT  (this  was  required  by  the  facility  for  the  feedback  loop). 


The  initial  tests  were  carried  out  with  a  power  spectral  density  of  the  applied  loading 
constant  over  the  100  Hz  to  500  Hz  range,  however  when  using  this  bandwidth  only 
155  dB  overall,  could  be  achieved.  In  an  attempt  to  increase  the  strain  levels  it  was 
decided  to  reduce  the  bandwidth  to  200  Hz.  The  bandwidth  (BW)  was  subsequently 
reduced  to  100  Hz  and  finally  1/3  octave  centred  on  the  predominant  response 
frequency  of  the  panel.  When  failure  occurred  a  dramatic  change  in  response  was 
observed.  The  failure  mechanism  was  that  of  core  failure  as  shown  in  Figure  4.0.  There 
was  no  indication  of  facing  or  backing  skin  distress. 


997 


5.0  Comparison  With  Theoretical  Predictions 
5.1  Fundamental  Frequency 

From  the  strain  gauge  readings  the  panel  was  seen  to  be  vibrating  with  simply 
supported  edge  conditions.  Soovere  [7]  suggests  that  "effective"  dimensions 
(essentially  from  the  start  of  the  pan  down)  be  used  to  determine  the  fundamental 
frequency  which  is  given  by; 


x\n 


This  equation  is  applicable  to  simply  supported  panels  with  isotropic  facing  and 
backing  skins,  thus  for  the  purpose  of  applying  the  above  equation,  the  actual  section 
was  approximated  to  a  symmetric  (isotropic)  section.  The  predicted  fundamental 
frequency  is  given  below.  It  was  observed  however,  that  if  the  panel  dimensions  are 
taken  relative  to  mid  way  between  the  staggered  pitch  of  the  fasteners  a  significant 
improvement  was  achieved  (see  "Soovere  (2)"  in  table  1).  Alternative  frequency 
estimations  using  an  FE  model  and  an  ESDU  data  item  [16]  are  summarised  in  the 
following  table; 


Method 

Freq.(Hz) 

%  Error 

Measured 

228 

- 

Soovere 

274.3 

+20.3 

Soovere 

(2) 

213.3 

-6.4 

FE 

239.04 

+4.8 

ESDU 

193 

-15.3 

Table  1  -  Comparison  of  calculated  frequencies  for  simply  supported  sandwich 

panel. 

Note;  the  percentage  error  is  based  on  the  actual  measured  response  frequency  of  the 
panel  in  the  PWT. 

Given  that  the  excitation  bandwidth  extended  (at  least  initially)  up  to  500  Hz,  modes 
up  to  500  Hz  were  obtained  from  the  FE  model.  In  actual  fact  2  FE  models  were  used, 
the  first  was  a  basic  model  with  380  elements,  however  a  more  detailed  model,  shown 
in  Figure  5.0,  (with  essentially  each  element  split  into  4)  was  used  for  the  results 
presented  in  this  paper.  The  predicted  modes  from  the  FE  model  were  as  follows; 


Mode 

No. 

Frequency 

(Hz) 

Mode  No.  in  x 
direction  (m) 

Mode  No.  in  y 
direction  (n) 

Figure 

No. 

1 

239.04 

1 

1 

6 

2 

334.0 

2 

1 

7 

3 

430.02 

1 

2 

8 

4 

1 

9 

Table  2  -  Finite  Element  Model  Predicted  Frequencies. 


998 


5.2  rms  Strain 

The  predicted  strains  were  calculated  using  Blevins'  normal  mode  method  (NMM)  [5], 
with  a  joint  acceptance  of  unity  for  the  fundamental  mode  of  vibration,  using  the 
following  expression; 


s^=  Lp.(fJ  .  ^  (2) 

In  an  attempt  to  improve  the  estimated  response,  the  rms  strain  was  calculated  for 
each  mode  within  the  bandwidth  of  excitation.  The  Joint  accetptance  for  each  mode 
was  calculated  using  equation  3  and  the  calculated  strains  for  each  mode  were  then 
factored  by  the  relevant  joint  acceltance  term.  The  overall  strain  was  then  calculated 
for  all  the  relevant  modes.  A  comparison  with  ESDI!  [16]  has  also  been  included, 
however  the  ESDU  method  does  not  provide  an  indication  of  shear  stress  in  the  core. 

Soovere  presents  a  simple  expression  for  the  joint  acceptance  function  for  a  simply 
supported  panel  excited  by  an  (acoustic)  progressive  wave,  for  the  case  where  n  is 
odd; 

•2  _  ^  (l-Cos(m7c)Cos(c0ra/c)  ,  . 

(1  -  (cD^a/ mrcc)^) 

Note,  when  n  is  even  the  joint  acceptance  is  zero. 

Given  that  the  bandwidth  varied  for  the  applied  loading,  the  overall  SPLs  were 
expressed  as  spectrum  levels  for  the  purpose  of  comparison  in  the  linearity  results,  the 
results  (both  measured  and  predicted)  have  been  summarised  in  table  3,  (SGI  results 
have  been  plotted  in  Figure  10.0).  The  results  from  the  ESDU  data  item  [16]  have 
been  included  in  table  4  for  comparison. 

Note  -  due  to  recorder  channel  limitations  SGI  &  SG2  were  not  connected  at  the  time 
of  failure  and  no  results  were  available  at  the  highest  sound  pressure  levels. 


OASPL 

(dB) 

Spectrum 
Level  SPL 
(dB) 

Measured 
Strains(u£) 
SGI  SG2 

Calculated 
(ps)  j=l 

SGI  SG2 

Calculated 
(Multi  Mode) 
SGI  (us)  SG2 

130 

107 

7 

7 

8.7 

9.2 

2.5 

2.6 

140 

117 

20 

19 

27.5 

29.1 

8.0 

8.1 

150 

127 

55 

60 

87.0 

92.0 

25.4 

25.8 

155 

132 

100 

100 

154.7 

163.7 

45.1 

45.8 

157 

134 

130 

130 

194.8 

206.0 

56.8 

57.6 

163 

140 

- 

- 

388.7 

411.1 

202.5 

162.7 

164 

141 

- 

- 

436.1 

461.2 

312.0 

250.7 

Table  3  -  Comparison  of  Measured  &  Predicted  rms  Strains 
for  the  Panel  Centre,  Facing  &  Backing  Skin  Gauges. 


999 


OASPL 

(dB) 

Spectrum 
Level  SPL 
(dB) 

Measured 
Strains  (pe) 
SGI  SG2 

ESDI!  Strains 

SGI  SG2 

130 

107 

7 

7 

10.1 

29 

140 

117 

20 

19 

150 

127 

55 

60 

101 

290 

155 

132 

100 

179.5 

515.7 

157 

134 

130 

■E9 

253.6 

728.5 

163 

140 

- 

- 

637 

1830 

164 

141 

- 

- 

1010 

2900 

Table  4  -  Comparison  of  Measured  &  ESDI!  Predictions  of  the 
rms  Strains  for  the  Panel  Centre,  Facing  &  Backing  Skin  Gauges. 

There  is  a  considerable  difference  in  the  calculated  response  from  using  a  joint 
acceptance  of  unity  for  the  fundamental  mode  and  that  when  estimating  the  joint 
acceptance  for  each  mode  and  calculating  the  overall  response  for  several  modes, 
however  it  was  observed  that  if  the  average  value  from  both  methods  is  used  the 
response  compares  favourably  with  that  measured  (-at  least  for  the  cases  under 
consideration).  The  average  value  has  been  included  on  the  linearity  plot  for  SGI, 
shown  in  Figure  10.0). In  general,  the  level  of  agreement  between  theory  and  practice 
was  considered  adequate  and  it  was  decided  to  apply  the  theory  to  estimating  the  shear 
stresses  in  the  core  (Table  5); 


OASPL 

(dB) 

Spectrum 
Level  SPL 
(dB) 

Core  Shear 
Stress  (J=l) 
(rms  psi) 

Core  Shear 
Stress 

(Multi  Mode) 
(rms  psi) 

Average 
Core  Shear 
Stress 
(rms  psi) 

Peak  Core 
Shear 
Stress 

_ (E£!i _ 

130 

107 

0.3 

0.09 

0.19 

0.58 

140 

117 

0.94 

0.28 

1.84 

150 

127 

2.99 

0.88 

1.93 

5.8 

155 

5.31 

1.57 

3.44 

10.32 

157 

134 

6.69 

1.97 

4.33 

13.0 

163 

140 

13.35 

5.57 

9.46 

28.38 

164 

141 

14.98 

8.59 

11.78 

35.34 

Table  5  -  Predicted  Core  Shear  Stress. 

6.0  Discussion  &  Recommendations 

The  ESDI!  method  proved  very  conservative  and  will  thus  give  a  degree  of 
confidence  when  used  in  the  early  stages  of  the  design  process.  Blevins  Normal  Mode 
Method  was  observed  to  give  reasonable  accuracy  in  predicting  the  highest  strains  in 
the  panel  and  would  merit  use  when  designs  have  been  fixed  to  some  degree;  at  which 
stage  FE  models  become  available. 

For  panels  whose  predominant  response  is  in  the  fundamental  mode  it  is  accepted  that 
the  contribution  from  shear  to  overall  deformation  is  very  small.  The  main  concern 
when  designing  a  honeycomb  sandwich  panel  which  is  subject  to  "severe"  acoustic 


1000 


loads  has  tended  to  focus  on  skin  strains  and  to  some  degree  the  properties  of  the  core 
material  have  been  ignored.  The  fact  that  low  skin  strains  are  observed  has  the  effect  of 
giving  an  impression  that  there  is  no  cause  for  concern,  however  when  the  properties 
of  the  core  material  are  low  or  unknown,  some  caution  is  required.  There  is 
unfortunately  no  available  S-N  data  for  the  type  of  honeycomb  used  in  the  construction 
of  the  panel,  however  the  allowable  ultimate  strength  for  the  core  material  is  of  the 
order  of  26  psi,  so  clearly  the  163  dB  level  was  sufficient  to  cause  a  static  failure  while 
the  lower  SPLs  can  be  assumed  to  the  have  contributed  to  initiating  fatigue  damage. 

On  cutting  up  the  test  panel,  a  large  disbond  was  observed  however  it  did  not  extend 
to  the  panel  edge  where  cracking  had  occurred  (the  mid  point  of  the  long  edge  being 
the  location  of  maximum  shear  for  a  simply  supported  panel)  and  it  was  the  opinion  of 
the  materials  department  that  the  failure  had  not  initiated  in  the  disbond. 

The  SPLs  used  in  the  test  were  not  excessively  high  and  were  comparable  to  service 
environments  (an  example  of  which  is  given  in  Table  6).  It  should  be  noted  that  while 
the  levels  in  Table  5  are  1/3  octave  bandwidths,  the  actual  spectrum  is  not  generally 
flat  within  each  band  for  engine  intakes,  but  is  rather  made  up  of  tones  (Figure  11). 
These  tones  or  spectrum  levels  can  thus  essentially  be  the  band  level  and  thus  some 
caution  should  be  exercised  when  converting  intake  band  levels  to  spectrum  levels 
using  the  traditional  approach  [17]. 


1/3  Octave  Centre 

Sound  Pressure 

Frequency  (Hz) 

Level  (dB) 

100 

141 

125 

133 

160 

140 

200 

142 

250 

140 

315 

139 

Table  6  -  Typical  Acoustic  Service  Environment. 

Note;  Overall  levels  may  reach  160  -  170  dB,  however  they  tend  to  be  influenced  by 
SPLs  at  blade  passing  frequencies,  which  are  much  higher  than  panel  fundamental 
frequencies. 


7.0  Conclusion 

It  has  been  shown  that  although  moderate  levels  of  acoustic  excitation  produce  quite 
low  overall  rms  strains  in  the  skins  of  honeycomb  sandwich  panels,  it  is  still  possible, 
when  using  very  light  weight  cores,  to  generate  core  shear  stresses  of  a  similar  order  of 
magnitude  to  the  allowable  ultimate  strength  of  the  material. 


Acknowledgements 

The  author  acknowledges  the  support  of  Short  Bros.  PLC  in  the  course  of  preparing 
this  paper  and  also  the  assistance  of  Mr  Neil  McWilliam  with  regard  to  the  FE 
modelling. 


1001 


References 


I .  0  Air  worthiness  Requirements  (JAR/FAR)  Section  25 .57 1  .d. 

2.0  Miles,  J.W.,  "On  Stmctural  Fatigue  Under  Random  Loading, "  Journal  of  the 
Aeronautical  Sciences,  (1954),Vol.21,  p753  -  762. 

3.0  Powell,  A.,  "On  the  Fatigue  Failure  of  Structures  due  to  Vibrations  Excited  by 
Random  Pressure  Fields,”  Journal  of  the  Acoustical  Society  of  America,  (1958), 
Vol.30,  No.l2,pll30-  1135. 

4.0  Clarkson,  B.L.,  "Stresses  in  Skin  Panels  Subjected  to  Random  Acoustic 
Loading,"  Journal  of  the  Royal  Aeronautical  Society,  (1968),  Vol.72, 
plOOO-  1010. 

5.0  Blevins,  R.D.,  "An  Approximate  Method  for  Sonic  Fatigue  Analysis  of 
Plates  &  Shells,"  Journal  of  Sound  &  Vibration,  (1989),  Vol.129,  No.l, 
p51-71. 

6.0  Holehouse,  I.,  "Sonic  Fatigue  Design  Techniques  for  Advanced  Composite 
Aircraft  Structures,"  AFWAL  TR  80-3019,(1980). 

7.0  Soovere,  J.,  "Random  Vibration  Analysis  of  Stiffened  Honeycomb  Panels  with 
Beveled  Edges,"  Journal  of  Aircraft,  (1986),  Vol.23,  No. 6,  p537-544. 

8.0  Acoustic  Fatigue  Design  Data  (Part  1),  AGARD-AG- 162-72,  (1972). 

9.0  ESDU  International,  London,  Series  on  Vibration  &  Acoustic  Fatigue. 

10.0  Richards,  E.J.,  Mead,  D.J., "Noise  and  Acoustic  Fatigue  in  Aeronautics,"  John 
Wiley  &  Sons,  New  York,  (1968). 

II. 0  Clarkson,  B.L.,  "A  Review  of  Sonic  Fatigue  Technology,"  NASA  CR  4587, 

(1995). 

12.0  Millar,  D.,  "Analysis  of  a  Honeycomb  Sandwich  Panel  Failure,"  M.Sc.  Thesis, 
University  of  Sheffield,  (1995). 

13.0  Holehouse,  L,  "Sonic  Fatigue  of  Aircraft  Structures  due  to  Jet  Engine  Fan 
Noise,"  Journal  of  Sound  &  Vibration,  (1971),  Vol.  17,  No. 3,  p287-298. 

14.0  Soovere,  J.,  "Correlation  of  Sonic  Fatigue  Failures  in  Large  Fan  Engine  Ducts 
with  Simplified  Theory,"  AGAEUD  CPI  13  (Symposium  on  Acoustic  Fatigue), 
(1972),  pi  1-1  -  11-13. 

15.0  Mead,  D.J.,  "Bond  Stresses  in  a  Randomly  Vibrating  Sandwich  Plate:  Single 
Mode  Theoiy,"  Journal  of  Sound  &  Vibration,  (1964),  Vol.l,  No. 3, 
p258-269. 

16.0  ESDU  Data  Item  86024  (ESDUpac  A8624),  "Estimation  of  RMS  Strain  in 
Laminated  Face  Plates  of  Simply  Supported  Sandwich  Panels  Subjected  to 
Random  Acoustic  Loading,"  Vol.  3  of  Vibration  &  Acoustic  Fatigue  Series. 

17.0  ESDU  Data  Item  66016,  "Bandwidth  Correction,"  Vol.  1  of  Vibration  & 
Acoustic  Fatigue  Series. 


1002 


Backing  Skin 


Figure  2.0  -  PWT  Test  Specimen. 


1003 


Microstrain  (dB) 


Figure  3.0  -  Response  of  Strain  Gauge  SGI  During  Sine  Sweep. 


Figure  4.0  -  Section  Through  Failure  Region  in  PWT  Panel. 


1004 


Figure  5.0  -  PWT  Panel  Finite  Element  Model. 


Figure  8.0  -  FE  Mode  1  (m==l,  n=2)  Figure  9.0  -  FE  Mode  2  (m=3,  n=l) 


1005 


Sound  Pressure  Level  (dB) 


Figure  10.0  -  Linearity  Plot  for  SGI  (Measured  &  NMM  Prediction). 


Figure  11.0  -  Typical  Spectral  Content  of  Intake  Duct  Sound 
Pressures  with  Equivalent  1/3  Octave  Levels  Superimposed. 


1006 


Time  Domain  Dynamic  Finite  Element  Modelling  in  Acoustic  Fatigue  Design 


Authors: 

P.  D.  Green 
Military  Aircraft 
British  Aerospace 
Warton 

A.  Killey 

Sowerby  Research  Centre 
British  Aerospace 
Filton 

Summary 

Advanced  Aircraft  are  expected  to  fly  in  increasingly  severe  and  varied  acoustic  environments. 
Improvements  are  needed  in  the  methods  used  to  design  aircraft  against  acoustic  fatigue.  Since  fatigue 
life  depends  strongly  on  the  magnitude  of  the  cyclic  stress  and  the  mean  stress,  it  is  important  to  be  able 
to  the  predict  the  dynamic  stress  response  of  an  aircraft  to  random  acoustic  loading  as  accurately  as 
possible. 

The  established  method  of  determining  fatigue  life  relies  on  linear  vibration  theory  and  assumes  that  the 
acoustic  pressure  is  fully  spatially  correlated  across  the  whole  structure.  The  technique  becomes 
increasingly  unsatisfactory  when  geometric  non-linearities  start  to  occur  at  lugh  noise  levels  and/or 
when  the  structure  is  significantly  curved.  Also  the  excitation  is  generally  not  in  phase  across  the  whole 
structure  because  of  complex  aerodynamic  effects. 

Recent  advances  in  finite  element  modelling,  combined  with  the  general  availability  of  extremely  fast 
supercomputers,  have  made  it  practical  to  carry  out  non-linear  random  vibration  response  predictions 
using  time  stepping  finite  element  (FE)  codes. 

Using  the  time  domain  Monte  Carlo  (TDMC)  technique  it  is  possible  to  model  multi-modal  vibrations 
of  stiffened  aircraft  panels  without  making  the  simplifying  assumptions  concerning  the  linearity  of  the 
response  and  the  characteristics  of  the  noise  excitation. 

The  technique  has  been  developed  initially  using  a  simple  flat  plate  model.  This  paper  presents  some  of 
the  results  obtained  during  the  course  of  this  work.  Also  described  are  the  results  of  a  study  of  the 
“snap-through”  behaviour  of  the  flat  plate,  using  time  domain  finite  element  analysis.  For  simplicity,  it 
was  assumed  that  the  dynamic  loading  was  fully  in  phase  across  the  plate. 


Introduction 

Aircraft  structures  basically  consist  of  thin,  generally  curved,  plates  attached  to  a  supporting  framework. 
During  flight  these  stiffen^  panels  are  subjected  to  a  combination  of  static  and  dynamic  aerodynamic 
loads.  On  some  aircraft  there  may  be  additional  quasi-static  thermal  loads  due  to  the  impingement  of  jet 
effluxes  in  some  areas.  Parts  of  advanced  short  take  of  and  landing  (ASTOVL)  .aircraft  may  be  required 
to  withstand  noise  levels  up  to  175dB  and  temperatures  up  to  200deg  C.  Under  these  conditions  the 
established  methods  of  dynamic  stress  analysis  for  acoustic  fatigue  design  are  inappropriate  and  cannot 
be  employed. 

British  Aerospace  (BAe),  Sowerby  Research  Centre  (SRC)  and  Military  Aircraft  (MA)  have  been 
developing  a  method  to  predict  the  stress/strain  response  of  aircraft  structures  in  these  extreme  loading 
situations.  The  primary  .consideration  has  been  the  requirement  to  create  an  acoustic  fatigue  design  tool 
for  dealing  with  combined  static  and  dynamic  loads,  including  thermally  generated  “quasi-static”  loads. 


1007 


The  resonant  response  of  thin  aircraft  structures  to  aeroacoustic  loading  is  generally  in  a  firequency 
range  which  implies  that,  if  defects  form,  they  will  quickly  grow.  Hence  to  be  conservative,  it  is 
generally  assumed  that  a  component  has  reached  its  life  when  it  is  possible  to  find  quite  small  defects 
by  non-destructive  evaluation  techniques.  Several  different  materials  and  construction  methods  are  used 
in  modern  aircraft  and  so  there  are  a  number  of  possible  failure  criteria.  In  the  case  of  metals,  it  is  the 
presence  of  cracks  larger  than  a  certain  size.  For  composites  it  can  be  the  occurrence  of  either  cracking 
or  delamination.  Degradation  due  to  the  presence  of  microcracks  may  be  monitored  by  measuring  the 
level  of  stiffness  reduction  which  has  taken  place. 

This  philosophy  simplifies  the  type  of  stress  analysis  needed,  because  it  is  not  necessary  to  model 
structures  with  defects  present.  Materials  can  be  assumed  to  have  simple  elastic  properties  which 
remain  unchanged  throughout  their  lives.  In  consequence,  it  is  necessary  to  know  the  fatigue  behaviour 
in  terms  of  a  direct  relationship  between  number  of  cycles  to  failure  and  the  magnitude  of  the  “nominal” 
cyclic  stress,  or  strain,  at  a  reference  location. 

If  considered  important  and  capable  of  satisfactory  treatment,  the  relationship  can  be  modified  to  take 
into  account  material  property  changes  due  to  the  development  of  very  small  defects  at  points  of  stress 
concentration.  For  example,  metal  plasticity  in  the  region  of  a  small  crack,  could  be  included  in  an 
analysis  of  the  stress  distribution  around  a  fastener  hole.  It  is  well  known  that  plasticity  reduces  the 
peaks  of  stress  which  are  predicted  at  defects  by  analysis  which  assumes  perfectly  elastic  material 
behaviour. 

The  technique  developed  at  BAe  for  modelling  high  acoustic  loads  combined  with  possible  thermal 
buckling  uses  the  time  domain  Monte  Carlo  (TDMC)  technique  together  with  finite  element  analysis  by 
proprietary  FE  codes.  Response  characteristics  are  predicted  directly  in  the  time  domain  using 
simulated  random  acoustic  loadings.  These  may  then  be  used  in  fatigue  life  estimations  which  employ 
cycle  counting  methods  such  as  Rainflow  counting.  It  is  now  practical  to  predict  the  vibrational 
response  of  stiffened  aircraft  panels  without  the  necessity  to  assume  a  linear  response,  and  without 
simplifying  the  spatial  and  temporal  representation  of  the  noise  excitation. 

Since  the  technique  uses  proprietary  finite  element  codes,  quite  large  and  complex  models  of  aircraft 
structure  can  be  analysed  in  a  single  run.  Standard  pre-  and  post-processor  techniques  are  available  to 
speed  up  generation  of  the  finite  element  mesh  and  to  display  the  stress/strain  results. 

The  initial  development  work  was  carried  out  by  modelling  the  random  vibration  of  a  flat  plate.  For 
fully  in-phase  random  loading  at  low  noise  levels  the  predicted  response  is  predominantly  single  mode 
and  at  fhe  frequency  calculated  by  linear  theory.  However,  as  the  decibel  level  is  increased,  the 
frequency  of  the  fundamental  rises  due  to  geometrically  non-linear  stiffening.  At  veiy  high  dB  levels  the 
predicted  response  becomes  multi-modal;  the  resonance  peaks  move  to  higher  and  higher  frequencies 
and  broaden. 

The  effect  of  static  loading  on  the  response  has  been  studied  as  part  of  these  investigations  to  assist  in 
the  validation  of  the  methodology  being  developed. 


Thermal  Effects 

In  some  flight  conditions  it  is  possible  for  a  panel  to  be  buckled  due  to  constrained  thermal  expansion 
and  also  be  subjected  to  very  noise  levels  at  the  same  time.  An  example  is  when  a  ASTOVL  aircraft 
hovers  close  to  the  ground  for  an  extended  period,  panels  which  are  initially  curved,  or  thermally 
buckled  panels  may  possibly  be  snapped  through  from  one  side  to  the  other  by  a  large  increase  in 
dynamic  pressure. 

“Snap-through”  can  be  potentially  damaging  to  the  structure  of  an  aircraft  if  it  occurs  persistently, 
because  the  process  is  associated  with  a  large  change  in  the  cyclic  bending  stress  present  at  the  edge  of 
a  stiffened  panel.  High  performance  aircraft  must  therefore  be  designed  so  that  snap-through  never 
occurs  in  practice. 


1008 


The  dynamic  response  of  curved  panels  or  buckled  flat  plates  is  difficult  to  predict  theoretically  because 
of  non-linear  effects.  The  established  acoustic  fatigue  design  techniques,  which  are  based  on  linear 
vibration  theory,  are  only  able  to  provide  approximate  predictions  of  the  loading  regimes  in  which 
particular  panels  might  be  expected  to  undergo  snap-through. 

The  TDMC  method  can  be  used  to  model  non-linear  multi-modal  vibrations  of  stiffened  aircraft  panels 
which  are  also  subject  to  quasi-static  stress.  In  particular  calculations  may  be  carried  out  in  the  post- 
buckling  regime. 

With  this  technique  simulated  random  dynamic  pressure  loading,  with  measured  or  otherwise  known 
spectral  characteristics  is  applied  to  a  curved,  or  post-buckled  panel  and  the  time  domain  response 
calculated.  The  magnitude  of  the  dynamic  loading  may  then  be  increased  until  persistent  snap-through 
is  observed  in  the  predicted  response.  This  gives  the  designer  the  ability  to  design  out  the  potential 
problem  by  systematically  altering  the  most  important  parameters  in  order  to  identify  the  critical 
regime. 

Fatigue  Life  Estimation 

Although  acoustic  fatigue  is  a  complex  phenomenon,  it  has  been  established  that  the  life  of  a  component 
mainly  depends  on  its  stress/strain  history.  The  most  important  factors  in  this  regard  are  the  magnitude 
and  frequency  of  the  cyclic  strain  and  the  mean  level  of  stress  at  the  likely  failure  points.  On  this  basis 
fatigue  life  can  be  estimated  by  carrying  out  the  three  stage  operation  illustrated  in  Figure  1. 


Stage  1:  Determine  Loads 

A  determination  of  static  design  loads  is  relatively  straightforward  compared  to  a  calculation  of  th.e  full 
temporal  and  spatial  dependence  of  the  aeroacoustic  pressure  on  a  military  jet  in  flight.  This  is  an 
enormous  task  in  computational  fluid  dynamics  (CFD).  Designers  have  to  rely  on  experimental  data 
which  can  come  from  measurements  on  existing  aircraft  or  from  scale  model  tests  of  jets,  for  example. 
Existing  databases  can  be  extrapolated  if  the  circumstances  are  similar.  Experimental  noise  data  is 
usually  in  the  form  of  power  spectral  density  curves  as  opposed  to  time  series  fluctuating  pressures,  but 
either  can  be  used,  depending  on  the  circumstances. 

Stage  2:  Calculate  Stresses 

The  technique  chosen  to  obtain  the  stresses  clearly  depends  on  how  much  knowledge  there  is  about  the 
expected  loads.  In  the  early  stages  of  design  analytic  techniques  would  be  used  to  establish  approximate 
sizes  and  stress  levels.  However,  later  on  when  the  design  is  nearly  completion,  finite  element  (FE) 
stress  analysis  can  be  used  to  model  the  effect  of  random  acoustic  loading  on  the  parts  of  the  skin  which 
are  likely  to  be  severely  affected.  These  calculations  would,  of  course,  be  done  including  the  effect  of 
attached  substructure. 

The  established  method  of  designing  against  acoustic  fatigue  uses  a  frequency  domain  technique  which 
relies  on  the  validity  of  linear  vibration  theory.  The  method  forms  the  basis  of  a  number  of  methodology 
documents  published  by  the  Engineering  Sciences  Data  Unit  (ESDU).  Whenever  there^^e  large  out-of- 
plane  deflections  the  frequency  domain  method  cannot  be  used  because  of  the  “geometric  non¬ 
linearity”.  From  a  strictly  theoretical  point  of  view  such  analyses  have  to  be  carried  out  in  the  time 
domain,  although  approximate  methods  are  applied  with  some  success. 

The  established  technique  produces  inaccurate  results  for  curved  panels,  buckled  panels  and  for  panels 
under  high  amplitude  vibration.  Geometric  non-linearity  usually  stiffens  a  structure  in  bending  so  there 
is  a  tendency  to  overestimate  the  stress  levels  using  the  frequency  domain  technique.  This  conservatism 
is  clearly  useful  from  the  point  of  view  of  safety,  but  it  can  lead  to  possible  “over-design”. 
Unfortunately  this  is  not  always  the  case  when  there  are  compressive  static  stresses  present.  The 
established  method  also  fails  if  the  phase  of  the  noise  varies  significantly  over  the  surface  of  the 
structure,  which  is  the  case  in  a  number  of  aeroacoustic  problems.  The  techniques  under  development 
are  designed  to  overcome  these  problems. 


1009 


Stage  3:  Estimate  Fatigue  Life 


In  cases  of  random  acoustic  loading  it  is  customary  to  assume  that  damage  accumulates  according  to  the 
linear  Miner's  rule.  Fatigue  life  is  determined  from  experimental  data  in  the  form  of  stress  (or  strain 
amplitude),  S,  versus  number  of  cycles  to  failure,  N.  If  a  number  of  cycles,  n,  of  stress/strain,  S,  occur  at 
a  level  of  stress/strain  where  N(S)  cycles  would  cause  failure  then  the  fractional  damage  done  by  the 
n(S)  cycles  is  n(S)/N(S). 

Various  methods  have  been  developed  for  obtaining  n(S)  from  the  stress  (strain)  response.  If  the 
excitation  is  stationary,  ergodic  and  the  response  is  narrow  band  random  then  the  function  n(S)  can  be 
shown  to  be  in  the  form  of  a  Rayleigh  distribution  and  the  damage  sum  can  be  evaluated  from  plots  of 
root  mean  square  stress  (or  strain)  against  number  of  cycles  to  failure.  If  the  statistics  of  the  response 
are  not  Gaussian  then  it  is  necessary  to  count  the  numbers  of  stress  cycles  from  the  time  domain 
response  and  use  constant  amplitude  S/N  curves.  It  is  now  widely  accepted  that  the  best  way  of 
counting  the  cycles  is  to  use  the  Rainflow  method,  [1]. 

The  Loading  Regime 

The  loads  on  an  aircraft  may  be  conveniently  divided  into  static  and  dynamic. 

Loads  which  vary  only  slowly  are: 

a)  Steady  Aerodynamic  Pressure  Loading, 

b)  In-Plane  Loads  transferred  from  “external  structure”,  and 

c)  Thermal  Loads  due  to  Constrained  Expansions. 

The  rapidly  varying  loads  are,  of  course,  the  aeroacoustic  pressure  fluctuations  which  originate  from 
any  form  of  unstable  gas  or  air  flow. 

This  division  is  central  to  the  methodology  which  has  been  developed  because  it  enables  the  modelling 
to  be  carried  out  in  two  distinct  phases.  The,  so  called,  static  loads  do  vary,  of  course,  but  the  idea  is  to 
separate  effects  which  occur  on  a  time  scale  of  seconds  from  the  more  rapidly  varying  acoustic 
phenomena.  The  aim  is  to  split  the  loads  so  that  the  quasi-static  effects  can  be  calculated  in  an  initial 
static  analysis  which  does  not  depend  on  a  particular  dynamic  loading  regime.  Any  aerodynamic 
pressure  may  be  divided  into  a  steady  part  and  a  fluctuating  part.  The  natural  place  to  make  the  cut-off 
is  at  IHz  which  means  that  epoch  times  for  TDMC  simulations  are  then  of  the  order  of  a  second.  The 
epoch  time  must  not  be  too  short  because  of  statistical  errors,  and  it  cannot  be  too  long  because  this 
would  invalidate  the  assumption  of  constant  quasi-static  loads.  In  practice,  there  is  another  constraint  on 
the  epoch  time.  The  number  of  finite  elements  in  the  model  coupled  with  the  premium  on  cpu  time 
places  an  obvious  limit  on  the  epoch  time. 

Comparison  of  the  Time  and  Frequency  Domain  Methods 

A  flow  chart  comparing  the  two  methods  is  given  as  Figure  2.  The  main  difference  between  the  two 
techniques  lies  in  the  representation  of  the  dynamic  loads.  The  FD  method  uses  rms  loadings  and 
spectral  characteristics,  whereas  the  TD  method  uses  the  full  time  series  loadings.  Gaussian  statistics 
are,  de  facto,  assumed  by  the  FD  method,  but  this  is  not  necessarily  the  case  with  the  TD  technique. 

Application  of  the  frequency  domain  method  requires  that  the  response  is  dominated  either  by  a  single 
mode  or  a  small  number  of  modes.  To  determine  whether  or  not  this  is  the  case  in  practice,  a  normal 
modes  analysis  must  be  followed  by  a  determination  of  the  amount  of  coupling  between  the  excitation 
and  each  mode.  This  can  be  determined  quite  accurately  even  if  there  is  a  certain  amount  of  potential 
non-linearity  by  computing  the  joint  acceptances  for  each  mode,  which  are  overlap  integrals  of  the 
mode  shape  functions  with  the  spatial  characteristics  of  the  excitation.  Normally  these  quantities  will  be 
dominated  by  a  few  of  the  low  order  modes.  If  there  is  significant  coupling  into  more  than  one  mode 
then  it  will  be  necessary  to  use  the  TD  method  instead. 

With  the  time  domain  technique  it  is  possible  to  represent  the  dynamic  loads  in  a  way  which  models  the 
convection  of  the  noise  field  across  the  structure.  Very  complicated  loadings  can  be  applied  to  large 


1010 


models  but  in  consequence  it  can  be  difficult  to  validate  the  results  obtained,  because  they  cannot  be 
checked  against  anything  other  than  test  data  which  is  itself  subject  to  confidence  levels.  In  addition  it 
must  be  remembered  that  the  TDMC  results  themselves  are  subject  to  statistical  variability.  Finally  it 
should  be  noted  that  TDMC  data  must  be  used  in  conjunction  with  constant  amplitude  endurance  data. 
Rms  fatigue  data  can  only  be  used  with  frequency  domain  results. 

Time  Domain  Finite  Element  Modelling 

Until  recently,  the  majority  of  finite  element  analyses  were  applied  to  static  loading  conditions  or  “low 
frequency  normal  modes  analysis”.  The  method  involves  the  use  of  an  implicit  code  to  invert  in  one 
operation,  a  single  stiffness  matrix,  which  can  be  very  large.  The  general  availability  of  extremely  fast 
super-computers  has  now  made  it  possible  to  carry  out  large  scale  non-linear  dynamic  finite  element 
modelling  using  explicit  FE  codes.  These  codes  use  very  similar  types  of  element  formulation  to  the 
implicit  ones,  e.g.  shells,  solids  and  bars,  but  the  solution  is  advanced  in  time  using  a  central  difference 
scheme. 

One  potentially  very  useful  capability  of  time  domain  modelling  is  the  application  of  acoustic  pressure 
loadings  which  vary  both  in  the  time  and  spatial  domains.  If  the  spectral  characteristics  are  known, 
either  from  test  or  from  other  modelling  it  is  possible  to  generate  samples  of  random  acoustic  noise  and 
apply  these  directly  to  the  finite  element  model  as  a  series  of  “load  curves”. 

The  technique  for  determining  time  series  noise  was  developed  by  Rice  [2]  and  Shinozuka  [3].  They 
showed  that  homogeneous  Gaussian  random  noise  can  be  generated  from  the  power  spectral  density  as 
a  sum  of  cosine  functions  with  different  frequencies  and  random  phase.  Noise  can  be  temporally  and 
spatially  correlated  noise  by  deriving  phase  differences  from  cross  spectral  functions  if  they  are  known. 

The  TDMC  method  can  be  quite  costly  in  terms  of  central  processor  unit  (cpu)  time  because  the 
solution  must  be  recalculated  at  each  point  in  time.  To  reduce  execution  times,  the  explicit  codes 
employ  reduced  numbers  of  volume  integration  points  in  the  finite  element  formulations.  However  in 
this  work  cpu  times  are  extended  because  long  epoch  times  are  required  to  ensure  adequate  statistics.  It 
can  take  more  than  24hrs  to  obtain  a  solution  over  a  half  second  epoch  if  there  are  a  few  thousand 
elements  in  the  model. 

Hence  there  is  always  a  practical  limit  to  the  size  of  a  particular  time  domain  finite  element  analysis, 
(TDFEA).  If  the  loading  and  geometry  are  not  too  complicated,  the  frequency  domain  method  of 
analysis  can  be  tried  initially  to  gain  more  understanding  of  the  nature  of  the  response  in  an 
approximate  way.  In  some  cases  the  vibrational  response  regime  must  be  considered  carefully  to  decide 
whether  TDFEA  is  really  necessary.  These  may  be  situations  where  the  non-linear  effects  are  only 
moderate. 

It  would  be  ideal  if  the  full  dynamic  response  of  an  aircraft  could  be  determined  with  a  fine  mesh  model 
in  one  huge  operation,  but  experience  has  shown  that  this  requires  too  many  elements.  It  is  possible  to 
construct  frill  models  with  reduced  stiffness  using  superelements,  enabling  flutter  and  buffet  to  be 
studied,  because  these  are  essentially  low  frequency  phenomena.  However,  in  time  domain  analysis  it 
has  been  found  that  models  containing  a  large  amount  of  detail,  such  as  fasteners  and  individual 
composite  material  plys,  require  a  great  deal  of  cpu  time.  To  progress  we  must  devise  some  strategies  to 
overcome  this  situation.  Since  a  full  TDFEA  can  only  handle  a  part  of  the  aircraft  structure,  it  is  very 
important  that  loads  external  to  the  area  under  consideration  are  properly  taken  into  account.  This  is 
cmcial  to  the  success  of  this  type  of  modelling  as  it  is  to  all  finite  element  modelling. 

The  most  important  parameter  in  any  time  series  analysis  is  the  time  step.  This  is  determined  by  the 
velocity  of  sound  waves  in  the  structural  material,  and  is  generally  of  the  order  of  a/v  where  a  is  the 
shortest  element  dimension  and  v  is  the  velocity  of  longitudinal  sound.  A  small  time  step  is  therefore 
required  when  the  elements  are  small  and  the  velocity  of  sound  is  large.  For  an  aluminium  model  with 
10mm  square  elements  the  time  step  is  about  1.6ixs.  Hence  a  TDMC  run  with  a  half  second  epoch  time 
needs  about  a  half  a  million  steps.  A  simple  5000  shell  element  calculation  on  a  Cray  C94  would  take 
approximately  10  hours. 


1011 


Dynamic  FE  models  of  aircraft  structure  can  be  constructed  in  many  ways,  using  shell  elements,  beam 
elements  and/or  solid  elements.  Special  elements  exist  for  damping  and  for  sliding  interfaces.  Joints  can 
be  modelled  with  sliding  interfaces,  or  with  short  beams,  or  just  with  tied  nodes.  Fasteners  can  be 
modelled  with  small  solid  elements,  with  short  beams  or  with  tied  nodes,  also.  Unfortunately,  however 
short  beams  and  small  solid  elements  cause  a  dramatic  lowering  of  thC  time  step.  For  example,  if  the 
smallest  fastener  dimension  is,  say  3mm,  the  time  step  will  have  to  be  reduced  to  about  O.Sfis  if  solid 
elements  are  used  in  the  model.  The  effect  on  cpu  time  is  such  as  to  make  the  calculations  impractical. 
Sliding  interfaces  are  an  efficient  way  to  model  skin/substructure  contact  in  explicit  analyses,  but  it  is 
important  to  choose  the  algorithm  carefully  because  some  techniques  can  consume  large  amounts  of  cpu 
time. 

The  best  practical  way  of  representing  stiffened  aircraft  panels  for  TDMC  analyses  is  considered  to  be 
with  four  noded  shell  elements  simply  tied  together  at  their  edges.  A  number  of  efficient  shell 
formulations  are  avail-able  and  meshes  can  be  rapidly  produced  from  the  design  geometry.  Of  course, 
such  models  cannot  be  expected  to  produce  highly  accurate  stress  data  in  the  region  of  small  features 
but  this  aspect  has  to  be  sacrificed  in  the  interests  of  achieving  statistically  significant  amounts  of  time 
series  data.  To  improve  the  accuracy  of  stress  predictions  in  the  neighbourhood  of  stiffeners  etc.,  it  will 
be  necessary  to  couple  TDMC  analyses  with  fine  mesh  static  analyses. 


The  Generation  of  Time  Series  Data 

A  number  of  factors  must  be  borne  in  mind  when  generating  time  series  data  for  TDMC  calculations.  It 
is  important  to  consider  carefully  the  frequency  range  and  number  of  points  which  define  the  load 
spectrum  in  conjunction  with  the  epoch  time  and  number  of  points  on  the  time  series. 

The  Nyquist  Criterion  [4]  states  that  the  time  increment  must  be  less  than  or  equal  to  one  over  twice  the 
upper  frequency  on  the  power  spectral  density  curve.  For  the  sake  of  argument,  take  the  upper 
frequency  to  be  IkHz.  This  means  that  the  time  increment  must  be  less  than  500ps.  A  more 
conservative  time  increment  is  based  on  the  requirement  to  represent  the  dynamic  response  of  the 
structure  as  accurately  as  possible  over  a  full  cycle.  Assuming  a  resonant  frequency  of  500Hz,  which  is 
perhaps  near  the  limit  in  practice,  and  10  points  per  cycle  which  is  more  than  sufficient,  the  lower  limit 
on  the  time  step  works  out  at  about  2C)0|is.  Taking  all  these  factors  into  consideration,  the  number  of 
points  on  the  spectrum  curve  should  be  of  the  order  of  1000  and  there  should  be  between  1000  and 
5000  on  the  time  series.  Longer  epoch  times  can  be  used  but  for  reasons  of  practicality  and  statistics  it 
is  better  to  run  more  than  one  short  epoch  simulation  rather  than  one  long  simulation. 

Explicit  FE  modelling  frequently  requires  that  the  time  step  be  smaller  than  200jj,s.  In  the  example 
given  above  the  time  step  required  by  the  explicit  code  was  1.6fis.  Under  these  circumstances  the 
random  noise  could  be  defined  with  a  smaller  time  increment,  but  going  to  this  level  of  effort  has  been 
found  to  produce  no  measurable  change  to  the  calculated  response. 

Static  Initialisation 

There  are  two  possible  ways  of  dealing  with  the  effect  of  static  loads  in  TDMC  modelling.  Firstly  the 
complete  analysis  can  be  carried  out  using  the  explicit  code.  To  do  this  it  is  necessary  to  apply  only  the 
static  loads  to  the  model  and  run  the  code  until  equilibrium  is  reached.  By  introducing  a  high  level  of 
artificial  damping  the  stresses  created  can  be  relaxed  in  a  relatively  short  period  of  time.  The  time 
required  depends  on  the  lowest  resonant  frequency  of  the  structure  and  the  size  of  the  smallest  element 
in  the  model.  This  facility  is  termed  “dynamic  relaxation”. 

The  alternative  is  to  make  use  of  another  facility  in  the  explicit  code  called  “static  initialisation”.  The 
deformed  shape  and  stress  state  of  the  structure  with  just  the  static  loads  applied  are  first  obtained  very 
quickly  using  an  implicit  code.  The  solution  for  the  stressed  state  is  then  initialised  into  the  explicit 
code  prior  to  the  application  of  the  dynamic  loads.  Dynamic  relaxation  may  be  used  to  smooth  out  any 
differences  between  the  models. 


1012 


Damping  Representation 

Vibrating  aircraft  structures  are  damped  by  several  mechanisms,  for  example  friction  at  joints,  re¬ 
radiation  of  acoustic  waves,  and  energy  loss  in  viscoelastic  materials  It  is  difficult  to  generalise  about 
the  relative  importance  of  each  damping  process  in  practice.  Also  reliable  quantitative  data  is  not 
available  in  sufficient  detail  to  justify  the  inclusion  of  complex  models  of  damping  into  the  TDMC 
analyses.  Test  results  on  vibrating  stiffened  aluminium  panels  tend  to  show  that  the  damping  is  best 
approximated  by  a  combination  of  mass  and  stiffness  proportional  coefficients.  There  is  a  range  of 
frequencies  in  which  the  damping  ratio  can  be  considered  to  be  roughly  constant.  Until  more  detailed 
experimental  data  are  available  the  most  expedient  approach  is  to  assume  a  nominal  value  for  the  global 
damping  ratio  which  does  not  change  with  frequency.  Over  the  years  it  has  become  standard  practice  to 
assume  a  damping  ratio  of  about  2%  for  fastened  aluminium  structures. 

Equivalent  Linearisation 

There  are  some  loading  regimes  in  which  the  non-linear  response  to  high  levels  of  random  acoustic 
loading  can  be  approximately  found  using  a  linearisation  technique  combined  with  a  frequency  domain 
analysis.  The  basic  idea  is  to  replace  the  non-linear  stiffness  term  in  the  general  vibration  equation  by  a 
linear  term  such  that  the  difference  between  the  rms  response  of  the  two  equations  is  minimised  with 
respect  to  a  shifted  fundamental  resonance  frequency.  If  an  approximate  equation  for  the  non-linear 
stiffness  is  known  then  it  is  possible  to  derive  an  expression  for  the  shifted  “non-linear”  resonance 
frequency.  The  rms  response  to  random  acoustic  loading  may  then  be  found  by  combining  the 
Miles/Clarkson  equation  with  some  form  of  static  geometrically  non-linear  analysis.  References  to  this 
technique  are  Blevins  [5],  Mei  [6]  and  Roberts  &  Spanos  [7].  Where  the  geometry  is  complex  the  most 
appropriate  form  of  analysis  is  clearly  finite  element  analysis. 

Implementation  and  Validation  Studies 

The  stress  analysis  work  described  in  this  paper  has  been  undertaken  using  MSC-NASTRAN  and 
LLNL-DYNA. 

NASTRAN  is  a  well  known  implicit  finite  element  code  which  is  capable  of  handling  very  large 
numbers  of  elements.  It  has  been  developed  very  much  with  aerospace  structural  analysis  in  mind.  It  is 
basically  a  linear  analysis  code,  although  there  are  a  large  number  of  adaptations  to  deal  with  non-linear 
problems.  It  can  also  function  as  a  dynamic  code,  but  is  much  slower  than  DYNA  in  this  mode  because 
it  basically  needs  to  solve  the  complete  problem  at  each  time  step.  The  non-linear  features  which  are 
most  relevant  to  the  type  of  stress  analysis  being  discussed  here  are  those  concerned  with  geometric 
non-linearity.  Geometric  non-linearity  is  treated  by  dividing  the  load  into  a  series  of  steps,  obtaining  the 
solution  incrementally.  In  this  work,  the  code  has  been  used  for  linear  and  non-linear  static  analyses  and 
for  normal  modes  analyses. 

DYNA  is  an  explicit  finite  element  code  originally  developed  for  the  calculation  of  the  non-linear 
transient  response  of  three  dimensional  structures.  The  code  has  shell,  beam  and  solid  element  models 
and  there  are  a  large  number  of  non-linear  and/or  anisotropic  material  models  available.  DYTMA  was 
developed  primarily  for  the  modelling  of  impact  and  there  is  no  limit,  as  far  as  the  code  is  concerned, 
on  the  size  of  finite  element  model  which  can  be  analysed.  Many  of  its  advanced  features  relate  to 
impact  modelling  and  are  not  required  for  this  work.  One  useful  feature,  however  is  the  laminated 
composite  material  model  based  on  the  equivalent  single  layer  approximation.  This  code  has  been  used 
for  the  TDMC  calculations  presented  in  this  paper. 

Finite  element  models  for  the  stress  analyses  were  produced  using  MSC-PATRAN,  It  has  a  wide  range 
of  geometry  and  mesh  generation  tools  and  now  has  built  in  interfaces  for  both  NASTRAN  and  DYNA. 
The  element  definitions  are  compatible  with  both  codes  and  it  is  a  simple  matter  to  toggle  between  the 
two  codes  by  changing  the  analysis  preference.  Not  all  the  features  of  DYNA  are  supported  and  some 
of  the  parameters  must  be  set  by  editing  the  DYNA  bulk  data  produced  by  PATRAN. 

NASTRAN  results  were  post-processed  using  PATRAN.  DYNA  results  were  post  processed  using 
TAURUS,  which  is  faster  and  easier  to  use  than  PATRAN  for  this  task;  Some  special  in-house  codes 
have  been  written  to  generate  random  acoustic  noise  from  power  spectral  densities,  as  described  above, 


1013 


and  to  post  process  time  series  output  from  TAURUS.  One  of  the  codes  incorporates  a  fast  Fourier 
transform  (FTT)  routine  to  determine  spectral  responses  from  the  DYNA  time  series  predictions.  These 
codes  are  covered  under  the  generic  title  “NEW-DYNAMIC”. 

TDMC  Calculations  on  a  Simple  Flat  Plate 

Calculations  have  been  performed  on  a  very  simple  model  to  implement  the  TDMC  technique  and 
develop  the  in-house  software  referred  to  above.  A  PATRAN  database  was  constructed  representing  a 
simple  flat  plate,  350mm  x  280mm  x  1.2mm  thick  as  an  array  of  shell  elements  34x28.  For  simplicity 
the  boundary  conditions  were  taken  either  as  simply  supported  or  clamped.  There  are  a  number  of 
alternative  shell  element  formulations  available  in  DYNA,  [8].  The  Hughes-Liu  shell  was  used  initially 
because  of  its  good  reputation  for  accuracy,  but  later  a  switch  was  made  to  a  similar,  but  slightly  faster 
shell  element,  called  the  YASE.  It  was  found  that  equally  satisfactory  results  could  be  obtained  more 
quickly  using  this  element. 

Analyses  without  Static  Loads 

A  series  of  DYNA  calculations  were  carried  out  with  a  fiiUy  correlated  random  acoustic  pressure  load 
with  a  flat  noise  spectrum  between  OHz  and  1024Hz.  Investigations  were  carried  out  into  the  effect  of 
varying  the  sound  pressure  level,  the  epoch  time,  the  mesh  resolution,  the  damping  coefficient  and  the 
stochastic  function. 

Figure  3  shows  the  displacement  response  of  the  central  node  of  the  model  for  a  sound  pressure  level  of 
115dB  (about  12Pa  rms),  simply  supported  edges  and  mass  proportional  damping  set  so  that  the 
damping  ratio  was  equal  to  2%  at  the  fundamental  (1,1)  resonance  of  the  plate.  The  corresponding 
spectral  response  is  shown  in  Figure  4.  shows  a  sharp  resonance  peak  at  a  frequency  of  61.0Hz  which  is 
very  close  to  the  theoretical  frequency  of  the  (1,1)  mode  for  the  simply  supported  plate.  The  in-phase 
loading  means  that  only  the  modes  with  odd  numbered  indices  are  excited.  I^e  peaks  corresponding  to 
the  (3,1)  and  (1,3)  modes  are,  however,  not  visible  on  the  plot  because  they  are  too  small.  It  may  be 
concluded  from  these  results  that  the  behaviour  of  the  plate  at  these  pressure  level  is  well  within  the 
linear  regime. 

An  investigation  into  the  behaviour  of  the  rms  displacement  response  as  a  fijnction  of  SPL  was  carried 
out  by  increasing  the  loading  incrementally  from  75dB  (0.1 2Pa  rms)  to  175dB  (12kPa  rms).  The  results 
are  shown  in  Figure  5.  Also  shown  are  theoretical  predictions  obtained  using  the  Miles/Clarkson 
formula  with  NASTRAN  linear  and  non-linear  analyses  as  explained  above,  see  below  for  discussion. 

The  statistical  variation  of  the  results  was  investigated  by  repeating  a  half  second  epoch  TDMC  run  ten 
times  with  different  samples  of  flat  spectrum  noise.  It  was  found  that  the  standard  error  of  the  rms 
response  was  about  16%.  A  second  set  of  ten  repeats  were  carried  out  with  the  epoch  increased  to  2.5s. 
In  this  case  the  standard  error  reduced  to  roughly  8%.  From  the  theory  of  stochastic  processes,  it  can  be 
shown  that  the  standard  error  is  inversely  proportional  to  the  square  root  of  the  epoch  time.  On  this 
basis  therefore  the  ratio  between  the  standard  errors  should  be  equal  to  the  square  root  of  five,  or  2.23. 
From  the  analyses  this  ratio  is  about  2.  Further  runs  established  that  these  results  are  not  affected  by  the 
vibration  amplitude,  even  when  the  response  becomes  non-linear. 

Cautiously  therefore,  it  can  be  concluded  that  the  variance  of  the  TDMC  results  is  unaffected  by  non¬ 
linearity  of  the  response.  This  is  an  important  finding  because  it  builds  confidence  in  the  technique.  In 
many  practical  situations  it  may  be  necessary  to  rely  on  just  one  simulation  and  an  appropriate  factor  of 
safety.  It  can  be  quite  time  consuming  to  carry  out  a  large  number  of  repeat  TDMC  simulations.  The 
level  of  variance  would  be  first  established  by  repeating  one  load  case  a  number  of  times,  before 
confidently  applying  it  to  the  results  of  other  load  cases. 

Comparisons  with  Linearised  Theory 

The  linear  theory  of  plate  bending,  [9],  leads  to  relationships  between  the  central  deflection,  w,  of  a 
rectangular  plate  and  a  uniform  static  pressure  load,  Pstat  which  take  the  following  form. 

Psutab  =  kcffW  (1) 


1014 


where  a  and  b  are  the  length  and  breadth  of  the  plate,  and  k^ff  is  an  effective  stiffness  parameter  which  is 
a  function  of  the  modulus  of  rigidity  of  the  plate  and  the  edge  boundary  conditions.  For  the  plate 
studied  k^ff  is  about  30N/mm  for  the  case  of  simply  supported  edges  and  lOON/mm  for  clamped  edges. 
The  above  equation  only  holds,  however,  at  very  low  amplitudes,  as  can  be  seen  from  Figure  6.  This 
compares  geometrically  non-linear  NASTRAN  predictions  with  the  linear  ^eory.  Curves  are  shown  for 
both  simply  supported  and  clamped  boundary  conditions.  The  finite  element  results  show  the 
characteristic  hardening  spring  type  of  non-linearity. 

At  higher  amplitudes  the  dynamic  behaviour  may  be  approximately  predicted  using  “equivalent 
linearisation”  theory,  which  assumes  that  the  response  remains  predominantly  single  mode,  but  with  a 
resonant  frequency  which  rises  as  the  stiffness  of  the  structure  increases.  When  the  deflection  is  large 
the  static  force-deflection  relationship  can  be  written  as  the  sum  of  a  linear  stiffness  term  and  a  cubic 
non-linear  term: 

psiatab  =  kw(l+pw^)  (2) 

where  a  b  is  the  force,  k  is  the  linear  stiffness.  The  equation  is  written  with  the  leading  term  factored 
out  to  emphasis  the  point  that  p.  is  a  constant  which  is  small  compared  to  the  rms  deflection.  In  the  limit 
of  small  w  we  can  expect  the  pw^  term  in  the  brackets  to  be  negligible  compared  to  one,  which  means 
that  the  k  in  this  equation  must  be  the  same  as  kcff  above. 

Equation  2  was  fitted  to  the  NASTRAN  results  shown  in  Figure  6  to  find  the  best  fit  values  of  k  and  p. 
Table  1  shows  the  results  compared  with  the  effective  stiffness  calculated  from  linear  plate  bending 
theory.  It  can  be  seen  that  the  theoretical  stiffness  is  almost  identical  to  the  best  fit  k  from  the  non-linear 
finite  element  analysis. 

With  reference  to  the  results  in  Figure  5,  it  is  obvious  that  the  nature  of  the  response  is  strongly 
dependent  on  the  amplitude  of  the  vibrations.  For  rms  displacements  up  to  about  4%  of  the  plate 
thickness  the  behaviour  was  completely  linear.  For  displacements  between  4%  and  150%  of  plate 
thickness,  the  response  was  essentially  single  mode  dominated  but  the  level  could  not  be  predicted  by 
the  Miles/Clarkson  approach.  The  “equivalently  linear”  solution  does,  however,  agree  with  the  DYNA 
result  up  to  a  displacement  of  about  l.8mm.  The  linearisation  approach  cannot  be  expected  to  be 
correct  for  displacements  above  about  1.5  times  plate  thickness.  Above  this  point  the  response  predicted 
by  DYNA  was  multi-modal  and  strongly  non-linear.  The  equivalently  linear  predictions  departed 
considerably  from  the  DYNA  results  when  the  vibration  amplitude  was  very  high. 

It  was  also  observed  that  the  frequency  response  peaks  became  increasingly  noisy  for  higher  pressures, 
representing  the  increased  level  of  non-linearity  in  the  plate  vibrations.  The  increase  in  the  frequency  of 
the  fundamental  mode  with  acoustic  pressure,  as  calculated  by  DYNA,  is  shown  in  Figure  7.  Predictions 
from  equivalent  linearisation  theory  and  from  the  theory  of  Duffing's  equation  are  also  included,  see 
Nayfeh  &  Mook  [10]. 


t„,  =  f(l  +  3nw™,Y’ 

Equiv.  Lin.  Pred. 

(3) 

f„,  =  f(l+(3/8)w„,^) 

Duffing's  Eq.  Pred. 

(4) 

The  DYNA  results  lie  mostly  between  the  two  theoretical  curves,  agreeing  particularly  well  with  the 
results  of  equivalent  linearisation  theory  up  to  around  700Pa  (151dB  rms).  The  level  of  agreernent 
obtained  shows  that  the  frequency  response  behaviour  of  the  DYNA  model  is  similar  the  theoretical 
predictions,  providing  an  independent  check  on  the  results.  As  might  be  expected,  at  around  700Pa  the 
agreement  begins  to  breakdown,  since  the  linearisation  theories  are  not  valid  for  deflections  which  are 
significantly  greater  than  the  plate  thickness.  It  may  be  concluded,  however,  that  the  effect  of  geometric 
non-linearity  at  high  amplitudes  is  being  computed  by  DYNA  in  a  reasonably  accurate  manner.  A 
detailed  comparison  with  experimental  data  is  needed  to  determine  the  accuracy  of  the  DYNA  response 
predictions  themselves. 


1015 


Due  to  the  increasingly  irregular  shape  of  the  frequency  response  functions  derived  from  the  DYNA 
time  series  predictions  at  high  acoustic  loads,  it  was  not  possible  to  calculate  very  accurate  peak  widths 
for  pressures  above  approximately  135dB  (I20Pa  rmsX  Figure  8  shows  that  the  width  of  the  peak 
increased  with  increasing  acoustic  pressure,  but  not  in  a  regular  manner.  When  the  damping  is  mass 
proportional,  equivalently  linear  theory  predicts  that  the  width  of  the  peak  should  remain  unchanged  as 
the  pressure  rises.  This  is  because  the  geometric  stiffening  effect  of  rising  acoustic  pressure  exactly 
cancels  the  effect  of  a  smaller  damping  ratio  at  the  higher  resonant  frequency.  This  graph  shows  this  as 
a  horizontal  straight  line  at  2.44Hz.  The  DYNA  result  is  closer  to  the  type  of  behaviour  observed 
experimentally  where  the  width  of  the  peak  generally  increases  with  increasing  the  sound  pressure  level. 

Analyses  with  Combined  Loads 

Further  work  was  conducted  with  static  loads  superimposed  on  different  levels  of  random  acoustic 
loading.  These  calculations  were  done  using  the  coupled  NASTRAN-DYNA  approach  outlined  earlier. 
That  is  to  say  the  deformed  geometry  was  obtained  by  applying  the  static  loads  to  a  NASTRAN  model, 
with  the  results  being  initialised  into  DYNA  and  dynamically  relaxed  before  the  dynamic  loading  was 
applied.  Calculations  were  performed  with  compressive  in-plane  loading,  static  pressure  loading  and 
thermal  loading.  With  the  exception  of  the  thermal  runs,  the  boundary  conditions  used  in  these  runs 
were  identical  to  clamped,  except  that  symmetrical  in-plane  movement  of  the  edges  was  permitted.  We 
have  called  these  conditions  “semi-clamped”.  It  has  been  found  that  the  fundamental  resonant  frequency 
of  the  plate  without  static  loading  is  only  reduced  by  a  very  small  amount  if  the  appropriate  in-plane 
degrees  of  freedom  are  released,  see  Figure  9.  These  boundary  conditions  are  actually  closer  to  those 
which  exist  in  reality  when  a  panel  in  built  into  a  larger  structure. 

Figures  10-14  show  results  of  some  of  the  analyses  which  have  been  carried  out.  They  give  time  series 
data  along  with  spectra  responses  calculated  by  the  in-house  post-processing  code.  Numerical  data 
derived  from  these  results  are  summarised  in  Tables  3-5. 

A  series  of  analyses  have  been  carried  out  with  compressive  in-plane  loads  equal  to  one  third  of  the 
theoretical  buckling  loads  in  compression.  For  the  plate  used,  the  forces  per  unit  side  length  were  - 
3.46N/mm  in  the  x-direction  and  -5.46N/mm  in  the  y-direction.  The  results  of  one  analysis  are  shown  in 
Figure  10.  It  has  been  found  that  the  response  remains  dominated  by  the  fundamental  (1,1)  mode  as 
long  as  the  plate  is  unbuckled  and  the  SPL  is  low.  The  softening  effect  of  the  compressive  loads  on  the 
frequency  agreed  quite  well  with  Rayleigh-Ritz  predictions,  [5],  up  to  an  SPL  of  ?dB.  At  higher  sound 
pressure  levels,  the  DYNA  results  reflected  stiffness  changes  which  were  greater  than  those  predicted 
by  the  theory.  The-  same  was  found  in  the  case  of  tensile  loading.  It  is  believed  that  these  differences  are 
due  to  approximations  built  into  both  the  Rayleigh-Ritz  theory  and  the  DYNA  code. 

Figure  1 1  shows  the  results  of  a  calculation  with  a  superimposed  normal  pressure.  The  magnitude  of  the 
pressure,  700Pa,  was  chosen  so  as  to  provide  an  example  of  “post-buckled”  analysis.  This  size  of 
pressure  causes  the  plate  to  bow  out  in  the  centre  by  about  0.6mm.  It  is  well  known  that  in  the  post- 
buckling  regime  the  random  response  of  a  plate  depends  upon  the  magnitudes  of  both  the  static  and 
dynamic  loads.  In  this  case  the  static  loading  was  large  compared  to  the  applied  dynamic  loads  and 
“snap-through”  did  not  occur.  The  plate  simply  vibrated  about  its  statically  deflected  position  in  the 
fundamental  mode  with  a  slightly  increased  frequency. 

To  provide  a  test  of  the  DYNA  thermal  stressing  capability,  and  to  carry  out  an  investigation  into 
“snap-through”,  several  analyses  were  carried  out  with  a  uniform  temperature  rise  of  lOdeg  C  applied  to 
the  plate  with  clamped  edges.  This  is  quite  sufficient  to  cause  buckling  because  the  resulting 
compressive  biaxial  stress,  c,  is  well  above  the  buckling  level,  Gb-  If  f  is  the  frequency  of  the 
fundamental  and  J  is  a  constant  equal  to  1.248  because  of  the  clamped  boundary  condition,  the  two 
stresses  can  be  determined  approximately  from 


a  =  EaT/(l-v^) 

(5) 

Cb  =  4pa^f2/J 

(6) 

1016 


where  E,  a,  v  and  p  are  Young's  modulus,  coefficient  of  thermal  expansion,  Poisson's  ratio,  and  density 
respectively.  Using  these  formulae  we  find  c  =  24MPa  and  Ob  =  14MPa. 

Analyses  were  carried  out  with  several  different  levels  of  dynamic  load.  The  results  of  three  of  the 
calculations  are  shown  in  Figures  12,13  and  14.  It  was  found  that  the  threshold  for  snap-through 
occurred  at  an  acoustic  load  of  about  IkPa,  see  Table  4.  Below  this  level  the  mean  deflection,  w,  is  a 
function  of  the  static  load  alone,  equal  to  about  2.8nun  (the  negative  sign  indicates  that  the  plate  has 
bowed  in  direction  of  negative  z).  At  higher  SPLs  the  mean  deflection  reduces  because  the  plate  snaps 
backwards  and  forwards  between  positive  and  negative  z.  The  calculated  response  spectra  for  these 
higher  level  runs,  show  an  additional  peak  at  a  very  low  frequency,  ie  less  than  lOHz.  This  is  an  artifact 
caused  by  the  snap-through  since  the  fundamental  resonance  of  the  clamped  plate  is  at  1 13Hz. 

Figures  10-14  all  show  probability  density  functions  derived  from  the  time  series  data.  The  fluctuations 
on  these  plots  are  caused  by  the  smallness  of  the  epoch  time.  In  all  cases,  except  for  the  thermal 
calculations  with  the  two  largest  acoustic  loads,  it  can  be  seen  that  the  functions  are  basically  Gaussian 
in  shape.  It  may  therefore  be  concluded  that  it  is  reasonable  to  assume  that  the  response  of  a  plate  in  the 
post-buckled  region  is  Gaussian  unless  there  is  a  large  amount  of  snap-through. 

Discussion 

The  work  described  is  the  starting  point  for  investigations  and  validations  using  more  complex  FE 
models.  Further  work  has  been  carried  out  using  models  including  curvature,  sub-structure  and  detailed 
features.  It  is  difficult  to  validate  the  predictions  obtained  from  such  models  by  comparing  with  test  data 
because  the  results  themselves  are  open  to  interpretation.  It  has  been  found  that  the  predicted  stress 
levels  are  closer  to  the  test  results  when  the  chosen  location  is  away  from  any  small  features.  The  lack 
of  good  agreement  in  the  neighbourhood  of  the  features  can  be  explained  by  the  relatively  coarse  mesh 
used  in  the  dynamic  models.  The  overall  level  agreement  was  much  better  than  that  between  predictions 
based  on  linear  or  equivalently  linear  theory  and  test.  On  the  basis  of  experience,  the  latter  tend  to  over¬ 
predict  by  upwards  of  factors  of  two  and  three.  From  this  work  it  has  been  found  that  the  DYNA 
predictions  tend  to  be  greater  than  test  by  amounts  which  vary  but  are  generally  much  less.  The  average 
over-prediction  was  about  40%  with  a  significant  change  as  a  function  of  location. 

TDMC  runs  can  take  a  significant  amount  of  computer  time  to  carry  out  and  it  is  believed  that  to  make 
further  improvements  the  technique  should  be  combined  with  detailed  stressing  using  static  finite 
element  analysis.  Inaccurate  results  can  be  obtained  if  the  boundaries  of  the  part  of  the  structure  under 
analysis  are  not  properly  restrained.  In  the  case  of  models  of  aircraft  panels  this  may  significantly  affect 
the  resonant  frequencies  which  in  turn  affects  the  level  of  calculated  dynamic  stress.  In-plane  loads  on  a 
panel,  perhaps  due  to  thermal  stressing,  can  alter  the  fundamental  by  as  much  as  100-200Hz.  Looked  at 
from  a  theoretical  point  of  view,  the  only  way  to  solve  this  problem  is  to  construct  a  second,  coarse 
model  of  the  component,  along  with  some  of  its  surrounding  structure.  An  initial  calculation  can  then  be 
carried  out  with  this  model  in  order  to  obtain  the  loads  and  boundary  conditions  for  subsequent 
application  to  the  original  model. 

The  dynamic  phenomenon  of  “snap-through”  cannot  be  modelled  using  existing  methods  and  so  the 
TDMC  /  finite  element  technique  offers  the  engineer  a  way  to  determine  where  the  likely  regions  of 
unstable  vibration  are  located  in  circumstances  where  the  structure  is  complicated  by  attachments  etc. 


Conclusions 

This  paper  has  sought  to  explain  how  time  domain  finite  element  modelling  can  be  used  to  assist  in  the 
design  of  aircraft  against  acoustic  fatigue.  Although  the  technique  is  computationally  intensive,  it  does 
have  a  place  in  the  effort  to  understand  complex  vibrations,  such  as  the  response  of  structures  to 
spatially  correlated  jet  noise  excitations,  or  interactions  between  high  sound  pressure  levels  and  thermal 
loads. 

The  work  at  BAe  is  continuing  in  an  attempt  to  provide  the  analyst  with  a  greater  ability  to  determine 
dynamic  stress  levels  in  advanced  structures  with  complex  loadings. 


1017 


References 


1.  Dowling  N.  E.  ,  Fatigue  Prediction  for  Complicated  Stress  Strain  Histories,  J  Materials  1,  71 
(1972). 


2.  Rice,  In  Selected  Papers  on  Noise  and  Stochastic  Processes,  Ed  N  Wax  pplSO,  Dover  New  York 
(1954). 

3.  Shinozuka  M.  ,  Computers  and  Structures,  2,  855,  (1972). 

4.  Bendat  J.  S.  and  Piersol  A.G.  ,  Engineering  Applications  of  Correlation  and  Spectral  Analysis 
Wiley  (1990). 

5.  Blevins  R.D,  ,  An  approximate  method  for  sonic  fatigue  analysis  of  plates  and  shells,  J  Sound  and 
Vibration,  129,  1,  51  (1989). 

6.  Mei  C.  and  Paul  D.B.  ,  Non  Linear  multi-modal  response  of  a  clamped  rectangular  plates  to 
Acoustic  Loading,  AIAA  Journal,  24,  634,  (1986). 

7.  Roberts  J.  B.  &  Spanos  P.D  ,  Random  Vibration  and  Statistical  Linearisation,  Wiley,  (1990). 

8.  Whirley  R.G.  and  Engelmann  B.E.  ,  DYNA3D:  A  Nonlinear,  Explicit,  Three  Dimensional  Finite 
Element  Code  for  Solid  and  Structural  Mechanics  —  User  Manual,  Lawrence  Livermore  National 
Laboratory,  UCRL-MA- 107254  Rev.  1,  (Nov  1993). 

9.  Szilard  R. ,  Theory  and  Analysis  of  Plates,  Prentice  Hall,  New  Jersey. 

10.  Nayfeh  A.H.  &  Mook  D.T.  ,  Non  Linear  Oscillations,  Wiley  (1979). 


1018 


Parameter 

lUQgnnggll 

HISSH 

Linear  Theory 

NASTRAN 

NASTRAN 

Simply  Supported 

30.0 

30.7 

1.09 

Clamped 

104. 

101. 

0.266 

Table  1:  Values  of  Parameters  fitted  to  NASTRAN  results  compared  with  the 
linear  theory  values. 


Static  Loads 

Acoustic  Load 
Rrms 
(Pa/dB) 

Theory  ] 

1  DYNA 

(N.mm 

N, 

(N.mm 

Freq 

(Hu) 

Wfim 

(mm) 

■SI 

Wrms 

(mm) 

None 

1.2  (95.6) 

jnggi 

0.00290 

114 

0.01.2 

-3.46 

1.2  (95.6) 

0.00316 

94.1 

0.00313 

-3.46 

643.5  (150.1) 

1.69 

115 

1.08 

-3.46 

1.2  (95.6) 

68.33 

0.00359 

68.1 

0.00368 

-3.46 

700  (150.9) 

68.33 

2.09 

103 

1.42 

Table  2:  Summary  of  results  of  calculations  with  random  acoustic  loading 
superimposed  on  compressive  in-plane  loads. 


Static  Load 
p(Pa) 

Acoustic  Load 
(Pa/dB) 

DYNA 

None 

1.2  (95.6) 

114 

0.0000 

0.01.2 

700 

None 

— 

-0.646 

— 

700 

12  (115.6) 

115 

-0.655 

0.0189 

700 

700  (150.9) 

134 

-0.477 

1.28 

5  k 

None 

— 

-2.89 

— 

5  k 

12  (115.6) 

179 

-2.93 

0.0152 

5  k 

700  (150.9) 

173 

-2.70 

0.999 

Table  3:  Summary  of  results  of  calculations  with  random  acoustic  loading 
superimposed  on  static  pressure  loads. 


Temperature 
T  (<»C) 

Acoustic  Load 
(Pa  /  dB) 

1  DYNA  1 

None 

1.2  (95.6) 

114 

0.00256 

10 

None 

— 

-2.85 

— 

10 

1.2  (95.6) 

234 

-2.86 

0.0140 

10 

700  (150.9) 

219 

-2.80 

0.197 

10 

Ik  (154.0) 

9.01 

2.47 

1.02 

10 

1.2k  (155.6) 

5.01 

-0.717 

2.24 

10 

1.5k  (157.5) 

3.00 

-0,376 

2.20 

10 

2k  (160,0) 

92.2 

0.0185 

1.95 

10 

4k  (166.0) 

195 

0.0568 

2.08 

Table  4:  Summary  of  results  of  calculations  with  random  acoustic  loading 
superimposed  on  a  thermal  load. 


1019 


Figure  2:  Flowchart  Illustrating  the  Frequency  and  Time  Domain  Techniques 


(uiUi),ueujeoB,ds,a  ^  O 


Displacement  of  central  node  for  12Pa  rms  acoustic  pressure, 

1. 


DYNA 


Figure  4:  Spectral  Response  of  the  fiat  plate  corresponding  to  Fig  3.  Central 
Node  for  12Pa  rms  pressure,  DYNA  calculation. 


3 


2.5 


2 

1 

o 

^  1 


0.5 


0 

Figure  5:  Rms  Central  Deflection  of  the  plate  versus  sound  pressure  level 
Comparison  between  DYNA  results  and  linear  theory. 


3 


2.5 


2 

■£ 

£ 

E 

I 

I 

o 

1 


0.5 


0 

Figure  6:  Central  deflection  of  the  plate  versus  pressure,  NASTRAN  calcula¬ 
tions  compared  to  linear  theory. 


500  1000  1500  2000  2500  3000  3500 

Static  Pressure  (Pa) 


rms  Pressure  (Pa) 


1022 


Figure  7:  Variation  of  fundamental  frequency  of  the  (1,1)  mode  with  rms 
pressure,  Comparison  between  DYNA  and  theory. 


Damping  2%;  Stoctiastic  Fn  #1 ;  3Sx29  Nodes 

0  - 1 - 1 - 1 - 1 - - 

0  200  400  600  800  1000  1200 

rms  Pressure  (Pa) 

130  140  145  150  153  155 

SPL  (dB) 

Figure  8:  Variation  of  width  of  the  (1,1)  mode  resonance  peak  with  rms  pres¬ 
sure,  Comparison  between  DYNA  and  theory. 


1023 


Figure  9:  DYNA  model  predictions  for  random  vibration  of  the  plate  with 
semi-clamped  boundary  conditions. 


Figure  10:  Random  Vibration  results  with  compressive  load  in  the  y-direction 
of  -3.46N/mm  and  SPL  of  l.OPa. 


Figure  11:  Random  Acoustic  Loading  of  SPL=12Pa  superimposed  on  a  static 
pressure  of  TOOPa. 


1024 


Figure  12:  Random  acoustic  loading  of  SPL=700Pa  superimposed  on  a  thermal 
load  of  10  deg,  clamped  edges.  —  No  Snap  Through 


Figure  13:  Random  acoustic  loading  of  SPL— 2kPa  superimposed  on  a  thermal 
load  of  10  deg,  clamped  edges.  — Nearly  continuous  snap-through. 


Figure  14:  Random  acoustic  loading  of  SPL=4kPa  superimposed  on  a  thermal 
load  of  10  deg,  clamped  edges.  — Dominant  acoustic  load. 


1026 


SYSTEM  IDENTIFICATION  II 


ROBUST  SUBSYSTEM  ESTIMATION 
USING  ARMA-MODELLING 
IN  THE  FREQUENCY  DOMAIN 

by  U.  Prells,  A.  W.  Lees,  M.  1.  Friswell  and  M.  G.  Smart, 

Department  of  Mechanical  Engineering  of  the  University  of  Wales 

Swansea, 

Singleton  Park,  Swansea  SA2  8PP,  United  Kingdom 

ABSTRACT 

This  paper  reflects  early  results  of  the  research  on  modelling  the  influence  of 
the  foundation  on  the  dynamics  of  the  rotor.  The  foundation  is  connected  to 
the  rotor  via  journal  bearings.  Dynamic  models  exist  for  the  subsystems  of 
the  rotor  and  of  the  bearings;  the  first  is  reliable  but  the  latter  is  uncertain. 
The  foundation  model  is  unknown  and  has  to  be  estimated  using  rundown 
data.These  are  measured  responses  of  the  foundation  at  the  bearings  due 
to  unbalance  forces  of  the  rotor  which  are  assumed  to  be  known.  Uncer¬ 
tainties  in  the  bearing  model  will  be  transfered  to  the  estimated  foundation 
parameters.  The  main  scope  of  this  paper  is  to  introduce  a  method  which 
enables  the  decoupling  of  the  problem  of  model  estimation  and  the  problem 
of  the  influence  of  the  bearing  model  uncertainty. 

The  influence  of  changes  in  the  model  of  the  bearings  on  the  estimation 
of  the  foundation  model  is  mainly  due  to  the  sensitivity  of  the  computed 
forces  applied  to  the  foundation  at  the  bearings.  These  are  used  together 
with  the  associated  measured  responses  to  estimate  the  foundation  model  in 
the  frequency  domain.  Using  an  ARMA  model  in  the  frequency  domain  it  is 
possible  to  estimate  a  filtered  foundation  model  rather  than  the  foundation 
model  itself.  The  filter  is  defined  in  such  a  way  that  the  resulting  force  has 
minimum  sensitivity  with  respect  to  deviations  in  the  model  of  the  bearings. 
This  leads  to  a  robust  estimation  of  the  filtered  model  of  the  foundation. 
Since  the  filter  can  be  defined  in  terms  of  the  models  of  the  rotor  and  of  the 
bearings  only,  the  problems  of  estimating  the  foundation’s  influence  and  of 
the  sensitivity  of  the  estimates  with  respect  to  the  model  of  the  bearings 
are  decoupled. 

The  method  is  demonstrated  by  a  simple  example  of  a  single-shaft  rotor. 
Even  if  the  errors  in  the  bearing  model  are  about  50  %  the  relative  input  and 
output  errors  of  the  filtered  foundation  model  are  of  the  same  magnitude 
as  the  round-off  and  truncating  errors. 

1  INTRODUCTION 

An  important  part  of  a  machine  monitoring  system  for  fault  diagnostics  of 


1027 


a  turbo  generator  is  a  reliable  mathematical  model.  This  model  includes 
the  subsystems  of  the  rotor,  the  bearings  and  the  foundation.  The  model  of 
the  rotor  represents  the  most  reliable  knowledge,  the  model  of  the  journal 
bearings  is  uncertain,  and  despite  of  intensive  research  it  is  not  yet  possible 
to  define  a  model  for  the  foundation  which  refiects  the  dynamical  contribu¬ 
tion  to  the  rotor  with  sufficient  accuracy.  The  first  step  to  determine  the 
contribution  of  the  foundation  on  the  rotor’s  dynamic  performance  is  the 
estimation  of  a  reliable  foundation  model. 

Rundown  data  are  available,  i.e.  displacements  ufb{<->^)  ^  of  the 
foundation  at  the  bearings  which  are  due  to  an  unbalance  force  /c/(a;)  G 
of  the  m-shaft  rotor,  given  at  discrete  frequencies  a;  €  :=  {wi,  ■  •  • ,  wm}? 

and  this  data  may  be  used  to  estimate  the  foundation  model.  A  com¬ 
mon  method  ([1],[2],[3],[4],[5])  is  to  estimate  the  unknown  dynamic  stiffness 
matrix  6  i^4mx4m  foundation  at  the  bearings  using  the  in¬ 

put/output  equation 

F{uj)upb{(^)  =  (1) 

where  the  force  fpB  of  the  foundation  at  the  bearings  can  be  expressed 
by  dynamic  condensation  in  terms  of  the  data  ups,  fu  and  in  terms  of  the 
dynamic  stiffness  matrices  Ar,B  of  the  rotor  and  the  bearings  respectively 
yielding 

fpB  =  -Bufb  +  [0,  B]A~^q  ^  ^  .  (2) 

Here  the  dynamic  stiffness  matrix  Arb  of  the  rotor  mounted  on  the  bearings 
is  partitioned  with  respect  to  the  n  inner  degrees-of-freedom  (dof)  of  the 
rotor  and  to  the  4m  connecting  (interface)  dof 


^RB 


Aru  Arib 
Arbi  Arbb  +  B 


(3) 


The  non-zero  components  of  the  force  fpi  G  (D”  in  eq.  (2)  of  the  inner  part 
of  the  rotor  are  the  components  of  the  unbalance  force  /y,  i.e.  introducing 
the  control  matrix  Su  €  ^  dynamic  stiffness 

matrix  Ar  of  the  rotor  is  given  in  terms  of  the  matrices  of  inertia  and 
stiffness  which  are  defined  by  modal  analysis  and  by  its  physical  data  given 
by  the  manufacturer.  Each  of  the  m  shafts  of  the  rotor  is  connected  to 
the  foundation  usually  via  2  journal  bearings.  Since  the  dynamic  stiffness 
matrix  B  of  the  journal  bearings  represents  a  model  for  the  oil  film  it  consists 
of  connecting  dof  only.  It  can  be  shown  that  B  is  block  diagonal 


B  = 


0 


0 


(4) 


containing  the  dynamic  stiffness  matrices  Bi  =  Ki+  jujDi.i  =  1,  •  •  •  ,m,  of 
the  m  bearings.  Ki,  Di  are  the  matrices  of  stiffness  and  damping  respectively 


1028 


which  result  from  linearisation  and  are  in  general  non-symmetric  and  non¬ 
singular.  Eq.  (1)  is  then  used  to  estimate  the  foundation  transfer  function 
F{u).  This  has  been  discussed  in  several  papers  ([1],[2],[3],[4],[5]).  Lees  et 
al.  [3]  pointed  out  that  fpB  is  sensitive  with  respect  to  deviations  in  the 
model  of  the  bearings  over  part  of  the  frequency  range.  This  sensitivity  is 
transfered  to  the  model  estimates. 

In  this  paper  a  method  is  introduced  which  enables  the  decoupling  of  the 
two  problems  of  model  estimation  and  of  sensitivity  of  the  foundation  model 
with  respect  to  the  model  of  the  bearings.  The  basic  idea  of  this  method  is 
to  estimate  a  transfer  function  H{u)  which  maps  the  displacements  ufb{^) 
to  a  force  /^(w)  rather  than  the  force  i-e. 

H(u)ufb{^)  =  (5) 

In  extension  of  the  earlier  method  the  force  //f(a;)  can  be  chosen  to  be  of 
minimum  sensitivity  with  respect  to  the  model  of  the  bearings.  This  robust 
estimated  transfer  function  H{uj)  is  related  to  that  of  the  foundation  F(uj) 
by  a  transformation  P{u) 


H{lo)  =  PMF(a;),  (6) 

which  of  course  retains  the  sensitivity  with  respect  to  the  model  of  the 
bearings.  But  since  P{uj)  only  depends  on  the  models  of  the  rotor  and  the 
bearings  in  the  case  of  a  modification  within  the  model  of  the  bearings  no 
new  model  estimation  has  to  be  performed  because  this  has  been  done  ro¬ 
bustly  with  respect  to  such  model  changes. 


2  THE  OPTIMUM  CHOICE  OF  THE 
FORCE  VECTOR 

As  stated  in  Lees  et  al.  [3]  the  sensitivity  of  the  force  /fb  with  respect  to  B 
is  mainly  due  to  the  inversion  of  the  matrix  Arb  in  eq.  (2).  It  can  be  shown 
that  the  condensation  method  of  estimating  the  force  /fb  results  from  the 
special  case  of  eliminating  the  last  4m  rows  of  the  matrix 


W  := 


Arii 

^RBI 

0 


Arib 

ArbB  -b  P 


^  ^(n+8m.)x(n+4m) 


(7) 


which  can  be  written  as  Arb  =  T'^W  ^  (l^(n+4m.)x(n+4m)  defining  the 
selecting  matrix  of  the  master  dof  as 

T  :=  [ei,  •  •  •  ,  €  R(-+8n)x(n+4m)^  (S) 


1029 


where  in  general  en  denotes  a  unit  vector  of  appropriate  dimension  contain¬ 
ing  zeros  everywhere  but  in  the  nth  place.  In  extension  to  the  force  fpB 
defined  in  eq.  (2)  for  an  arbitrary  selecting  matrix  T  €  the 

condensation  leads  to  a  force  fn  given  by 


j-^LT 


In+Sm-W{T'^Wy'T^ 

”  *■  V*'  . . . 

=:P 


( 

Bufb 

V  } 


(  ^  \ 
0 

V  -^4m  J 


fpB- 


(9) 


Here  T-^  €  denotes  the  matrix  which  selects  the  slave  dof, 

and  in  general  In  denotes  the  unit  matrix  of  dimension  n.  Indeed,  inserting 
the  special  choice  of  T  from  eq.  (8)  into  eq.  (9)  leads  to  the  sensitive  force 
Ih  =  fpB  as  defined  in  eq,  (2). 

The  reason  for  the  sensitivity  of  /fb  is  that  the  subsystem  of  the  rotor 
has  low  damping.  Near  the  resonance  frequencies  of  Arb  its  large  condition 
number  depends  sensitively  on  B.  Thus  the  sensitivity  of  Jfb  with  respect 
to  B  is  due  to  a  large  condition  number  of  Arb  •  Let  T  denote  the  set  of  all 
possible  selecting  matrices,  i.e. 

r  :=  {[ei..--',ew4„]:e4elR"+*“. 

l<4<n  +  8m,  V  /c  =  1,  •  • « ,n  + 4m},  (10) 

One  criterion  for  an  optimum  choice  of  the  force  Jr  may  be  formulated  as 
the  following  minimisation  problem: 

Criterion  1: 

The  optimum  choice  is  the  solution  of 

mmcond{T^W),  (11) 

where  W  is  defined  in  eg.  (7). 

A  low  condition  number  is  necessary  but  not  sufficient  in  order  to  provide 
a  low  sensitivity  of  the  force  /h*  Therefore  a  numerical  test  can  be  applied 
using  stochastic  deviations  in  the  bearing  model.  Let  A  A  consist  of 

uniform  distributed  non-correlated  random  numbers  with  zero  mean  values 
and  variances  equal  to  1/3  for  alH  G  1,  ■  •  * ,  2m.  Define 

ABi  =  ABfisi,  Ti)  :=  siAKi  -h  A  A,  (12) 

where  the  positive  scalars  si.Ti  control  the  magnitude  of  the  random  error 
of  the  i-th  bearing  model.  Thus,  the  error  AB  =  AB{s,r)  of  the  bear¬ 
ing  model  is  well  defined  for  s  :=  (si,  •  ■  • ,  52771)"^  and  r  :=  (ri,  •  •  •  ,r2mV ■ 


1030 


Regard  the  force  fn  =  f{uj,T,AB)  as  a  function  of  the  selecting  ma¬ 
trix  T  and  the  bearing  model  error  AB.  For  I  random  samples  AB{k)  = 
AB(r{k),s{k)),k  ==  calculate  for  each  frequency  a;  G  the  up¬ 

per  and  lower  bounds  for  the  real  and  imaginary  part  of  each  component 
/i,  ^  =  1,  •  •  • ,  4m  of  the  force  vector  /,  i.e.: 


//Lax(“.U 

;=  max  Re  {/i(a;,T,  AB(/c))}  , 

(13) 

:=  min  Re  {fi{u,T,  AB{k))}  , 

(14) 

:=  ^max^Im  {/i(a;,T,  AB(/c))}  , 

(15) 

fLni^.T) 

:=  min  Im  AB{k))}  . 

k=l,"-,l 

(16) 

Defining  the  force  vectors 

/max(w,T) 

■■=  T)+j-  T), 

(17) 

/mm(w,T) 

(18) 

the  second  criterion  can  be  formulated  as  a  minimax  problem: 

Criterion  2: 

The  optimum  selection  is  obtained  from 

4m 

minmax^l/imax(w,r)  -  (19) 

TST  uj£u  .  - 
2  =  1 

Before  the  method  outlined  is  demonstrated  by  an  example  some  aspects  of 
the  mathematical  model  of  the  foundation  and  methods  for  its  estimation 
based  on  the  input/output  equation  (5)  will  now  be  considered. 


3  ESTIMATION  OF  THE  FOUNDATION 

MODEL 


The  purpose  of  this  section  is  to  estimate  the  unknown  foundation  model 
represented  by  the  matrix 

F{uj)  =  Afbb{^)  -  Afbi{^)Af]j{uj)Afib{^)‘  (20) 


This  expression  results  from  dynamic  condensation  of  the  dynamic  stiffness 
matrix  of  the  foundation 


Af 


Afbb  Afib 
Afib  Afh 


(21) 


1031 


which  is  partitioned  with  respect  to  its  inner  dof  (index  I)  and  those  dof 
coupled  to  the  bearings  (index  B).  For  viscous  damped  linear  elastomechan- 
ical  models  the  dynamic  stiffness  matrix  Aj?(a;)  of  the  foundation  is  given 
by  ^ 

Af(uj)  :=  (22) 

i=0 

The  matrices  Ai  are  real  valued  and  represent  the  contributions  of  stijffness, 
damping  and  inertia  for  z  =  0, 1, 2  respectively.  In  this  case  the  identifica¬ 
tion  of  the  foundation  model  requires  the  estimation  of  the  three  matrices 
Ai  which  are  parameterised  by  introducing  dimensionless  adjustment  pa¬ 
rameters  aik  €  IR,  for  all  /c  =  1,  *  •  • ,  ^  =  0, 1, 2  (see  for  instance  [6]  or 

[7] ).  Those  parameters  are  related  to  given  real- valued  matrices  Sik  by 

Ni 

Ai{ai) 

Jk=i 

Writing  the  adjustment  parameters  as  one  vector  a"’’  ;=  (af,aj’,aj)  G  IR^, 
p  :=  ATq  +  Ni  +  W2,  the  estimation  of  the  foundation  model  is  equivalent  to 
the  estimation  of  the  parameter  vector  a.  The  dynamic  stiffness  matrix  of 
the  foundation  becomes  a  nonlinear  function  of  this  parameter  vector 

^"(0;)  =F[(jj^a)  —  Afbb[^-)0)  “  Afb7(ci;j a)Ap}j(a;,  a)Af’/5(a;,  a).  (24) 

Substituting  the  measured  quantities  for  upB  and  fu  into  eqs.  (5)  and 
(9)  the  parameter  vector  a  is  usually  estimated  by  minimising  some  norm  of 
the  difference  between  measured  and  calculated  quantities,  called  residuals 

[8] .  Using  equation  (5)  is  equivalent  to  the  input  residual  method.  Defining 
the  ith  partial  input  residual  as 

(25) 

where  the  dependency  on  the  model  parameters  a  of  the  input  vector  is 
defined  by 

f{u),  a)  :=  P{uj)F[u),  a)uFB(w),  (26) 

the  cost  function  to  be  minimised  is  given  by 

M 

JKa):=£^;}(i)W,(i)^/(i),  (27) 

i=l 

where  Wj{i)  represents  a  weighting  matrix  for  the  zth  partial  residual  and 
the  superscript  f  denotes  the  conjugate- transpose.  The  inverse  problem  (27) 
is  nonlinear  with  respect  to  the  parameters  to  be  estimated.  Thus,  there 
is  no  advantage  relative  to  the  output  residual  method.  Defining  the  ith 
partial  output  residual  as 

vo{i)  :=  u{ui,  a)  -  (28) 


1032 


where  the  model  output  is  defined  by 


(29) 


0 

0 

ApiiiuJ^a)  . 
(30) 


(31) 

Woii)  denotes  a  weighting  matrix  for  the  ith  partial  output  residual. 

Mathematical  modelling  is  always  purpose  orientated  [9].  In  the  case 
discussed  in  this  paper  the  purpose  is  to  estimate  the  influence  of  the  foun¬ 
dation  on  the  dynamics  of  the  rotor.  For  this  purpose,  no  physically  inter¬ 
pretable  model  is  necessary  in  order  to  model  this  influence.  In  the  next 
section  an  alternative  mathematical  model  is  introduced  which  leads  to  a 
linear  inverse  problem. 

3.1  ARMA  MODELLING  IN  THE  FREQUENCY 
DOMAIN:  THE  FILTER  MODEL 

Auto  Regressive  Moving  Average  models  are  well  developed  (see  for  in¬ 
stance  [10], [7])  in  order  to  simulate  dynamic  system  behaviour.  ARMA 
models  are  defined  in  the  time  domain  by 

-  lAt)  =  -  ^A^),  (32) 

i=0  i=0 

where  the  present  output  (state  or  displacement)  u{t)  due  to  the  present 
input  f{t)  depends  on  rio  past  outputs  and  on  rii  past  inputs. 

In  the  frequency  domain  eq.  (32)  leads  to  a  (frequency-)  filter  model  [7]. 
With  reference  to  eq.  (5)  it  has  the  form 


Su 

0 

0 

0 


u(w,a)  :=  [0,0,/2(m+i),0].4  ^(w,a) 
with  the  dynamic  stiffness  matrix  A  of  the  entire  model 
A{u,  a)  = 


Arii{u)  Arib{^^)  0 

Arbi{^)  Arbb{‘^) B{lo)  —B(u) 

0  —B{uj)  R(a;)  4- Afbs(^)  g) 

0  0  Afib{^jO,) 


the  cost  function  to  be  minimised  is 


M 


Jo(a)  :=E^oW^oW«o(i). 


The  output  and  input  powers  Uo^rii  respectively,  and  the  matrices 
(•SA:)fc=o,-,ni  are  called  filter^parameters  and  Jiave  to  be  esti¬ 
mated.  Of  course  the  minimum  of  det[A(a;)]  and  of  det[B(cj)]  correspond 
to  the  resonance  and  anti-resonance  frequencies  of  the  subsystem  of  the 
foundation  respectively. 

For  an  optimum  choice  of  P  (see  eq.  (9))  the  estimation  of  A  and  B  can 
be  considered  to  be  independent  of  the  precise  values  of  the  model  of  the 
bearings.  Thus,  the  problem  of  the  uncertainty  in  the  bearmg  models  and 
the  problem  of  model  estimation  are  decoupled.  If  A  and  B  are  estimated 
refering  to  eq.  (5)  then 

=  H.  (34) 

The  estimation  of  the  filter  parameters  is  robust  with  respect  to  deviations 
in  the  bearings  model.  Thus,  the  uncertainty  of  the  estimation  of  the  foun¬ 
dation  model  F  is  due  to  the  inversion  of  the  matrix  P  only 

F  =  P-^H  =  p-'-B~^A,  (35) 

which  represents  a  problem  a  priori  and  which  occurs  only  in  the  calculation 
of  the  force  of  the  foundation  at  the  bearings 

/PB  =  P-^B~^Aufb-  (36) 

Of  course  the  force  vector  fpB  is  sensitive  to  changes  in  the  bearings  model 
but  only  due  to  corresponding  changes  in  P.  The  estimated  part  B  A  is 
robust  with  respect  to  changes  in  the  bearings  model. 

In  order  to  calculate  the  response  ufb  iio  explicit  calculation  of  the 
inverse  of  P  is  necessary, 

upB  =  A  Bfn-  (37) 

Since  the  estimated  model  and  the  force  Jh  are  insensitive  with  respect  to 
the  bearings  model  the  estimation  otupB  is  robust  in  this  sense. 

Of  course  the  influence  of  errors  in  up b  and  fu  have  not  yet  been  taken 
into  account.  Accordingly  the  model  powers  n^n^,  must  be  estimated  as 
well  as  the  matrices  Ai.Bk-  The  estimation  method  is  outlined  in  the  fol¬ 
lowing  section. 

3.2  ESTIMATION  OF  THE  FILTER 
PARAMETERS 

In  order  to  estimate  the  filter  parameters  the  least  squares  method  can  be 
applied  to  minimise  the  equation  error  in  eq.  (33).  Defining  the  zth  partial 
equation  residual  as 

vsij)  '=  A[ui)upB[^i)  -  B{uJi)fH{^i)  (38) 


1034 


the  cost  function  to  be  minimized  is  given  by 


M 

i=l 


(39) 


where  W^ii)  denotes  a  weighting  matrix  for  the  ith.  equation  residual.  As¬ 
suming  WeIi)  =  Am  for  alH  =  1,  •  •  ’ ,  M,  the  filter  equation  (33)  can  be 
extended  for  M  excitation  frequencies  as 

=  (40) 


where  C/,  2^  and  A  are  defined  by 


U 

Z 

A 


•  •  •  5  , 

r  a;i  0  1 


lom 


(41) 

(42) 

(43) 


The  solution  of  the  minimisation  problem  (39)  is  equivalent  to  the  normal 
solution  of  eq.  (40)  which  can  be  rewritten  as 


[Afio  1 ' ' '  1  -Ao )  ‘  :  ■^o] 


=:  V 


U 

Z 


=  0 


(44) 


Because  the  filter  parameters  represented  by  the  matrix  V  € 
j^4mx4m(no+ni+2)  leal-valued,  equation  (44)  must  be  satisfied  for  the 

real  and  imaginary  parts  of  the  matrix  Y  €  which  finally 

yields 

1/ [Re  {Y}  ,  Im  {Y}]  =:  VA  =  0.  (45) 

This  problem  does  not  lead  to  a  unique  solution  for  the  filter  parameters. 
Indeed,  for  any  arbitrary  non-singular  matrix  C 

CAufb  =  (46) 


is  also  a  solution.  But  since  one  is  interested  (see  eq.  (34))  in  the  product 
B~^  A  (or  its  inverse)  only  this  final  result  is  of  interest  and  this  product  is 
unique. 


1035 


As  a  necessary  and  a  sufficient  condition  for  a  full-rank  solution  V  of 
eq.  (45)  the  matrix  X  €  5^4m(no+ni4-2)x2M  a  rank  deficiency  of 

4m,  i.e. 

rank(A)  =  4m  (no  +  n^  +  1).  (47) 

Of  course  this  problem  has  to  be  treated  numerically.  The  rank  decision 
is  usually  made  by  looking  to  the  singular  values  7(no,  rii)  6  5^4m(no+ni+2) 
of  the  matrix  X  =  X{no,ni).  Because  one  cannot  expect  to  achieve  zero 
rather  than  relative  small  singular  values  one  has  to  define  a  cut-off  limit. 
This  is  due  to  the  fact  that  the  equation  error  (39)  can  be  made  arbitrary 
small  by  increasing  the  degree  p  :=  no  +  Ui  of  the  filter  model.  The  same 
situation  occurs  if  one  looks  at  the  maximum  relative  input  error 


6/  := 


max 


(48) 


or  to  the  maximum  relative  output  error 


en  :==  max 


II^fs(^2)1I 


(49) 


With  increasing  degree  p  the  errors  ej  and  cq  can  be  made  arbitrary  small. 

This  is  a  typical  expression  for  an  ill-posed  problem  which  can  be  turned 
into  an  well-posed  problem  by  applying  regularisation  methods  [11].  To 
choose  an  appropriate  regularisation  method  needs  further  investigation  and 
is  beyond  the  scope  of  this  paper.  In  the  next  section  the  method  of  choosing 
the  optimum  force  vector  fn  is  demonstrated  by  a  simple  example. 


4  A  SIMPLE  EXAMPLE 

The  test  model  is  depicted  in  Fig.  1.  The  one-shaft-rotor  is  simulated  by 
an  Euler-Bernoulli  beam  which  is  spatially  discretised  with  10  dof.  Accord¬ 
ing  to  the  partition  with  respect  to  inner  points  and  interface  points  (see 
eq.  (3))  the  number  n  of  inner  rotor  dof  is  8  and  the  number  of  connecting 
dof  is  2.  Only  one  translation  dof  of  the  rotor  is  connected  to  each  bearing 
which  are  modelled  by  massless  springs  with  stiffnesses  ki  —  1.77  •  10®  and 
k2  =  3-54  •  10®  N/m  respectively.  The  foundation  is  modelled  by  an  uncon¬ 
nected  pair  of  masses  mi  =  90,  m2  =  135  kg  and  springs  with  stiffnesses 
kfi  =  /c/2  =  1.77  •  10®  N/m.  The  force  fu  due  to  an  unbalance  6  =  0.01 
kg-m  is  given  by  fu{<^)  ■=  €  IR.  The  force  vector  f^i  e  IR®  is  assumed 

to  have  one  non-zero  component  only,  i.e.  Jri  fuc^.  The  frequency  range 
between  0  and  250  Hz  is  discretised  with  equally  spaced  stepsize  of  0.5  Hz. 
The  selecting  matrix  T  €  of  the  master  dof  is  assumed  to  consist  of 

the  unit  vector  64  in  order  to  select  out  the  unbalance  force  fjj  because  this 


1036 


Figure  1:  The  simple  test  model 

excitation  is  independent  of  the  model  of  the  bearings  and  therefore  of  min¬ 
imum  sensitivity.  Thus  the  remaining  redundancy  consists  in  eliminating 
one  row  of  the  matrix 

G  :=  [ei,  82. 83. 65.  •  •  • .  eul''  W  €  (50) 

The  result  of  the  first  criterion  are  depicted  in  Fig.  2.  It  shows  the  frequency 
dependent  condition  numbers  for  the  elimination  of  each  row  of  the  matrix 
G  in  turn.  This  leads  to  an  optimum  choice  by  eliminating  the  4-th  row  of 
G.  Thus,  the  optimum  choice  of  the  master  dof  is  given  by  T*^  =  [64,65]. 
This  result  is  confirmed  by  applying  the  second  optimisation  criterion. 

For  this  purpose  a  uniform  distributed  uncorrelated  random  error  with 
zero  mean  value  is  added  to  the  stiffnesses  of  the  bearings  simulating  a 
model  variation  of  50  per  cent,  i.e.  ki  — ^  ki  -1-  Nzikil2,  where  = 

1,2,  are  uncorrelated  random  numbers  with  expectation  value  E{Aki}  =  0 
and  with  variance  E{AkiAkj}  =  For  a  size  of  I  -  500  random 

samples,  the  upper  and  lower  bounds  /max, /min  and  /f  max, /h min  of  the 
force  fpB  =  /fb(<^,  A5)  and  of  the  force  Jh  =  /(w,T,  AB)  respectively 
with  the  selecting  matrix  T  -  [64,65],  have  been  calculated.  In  contrast  to 
the  maximum  difference  of  upper  and  lower  bounds  of  the  force  f^B  of  ~  20, 
that  of  the  force  fn  is  of  the  order  of  the  computational  accuracy  ~  10"'^, 
and  is  therefore  negligible.  In  a  first  step  the  force  /fb  is  used  to  estimate 
the  filter  parameters  of  the  model  F  of  the  foundation.  Solving  the  singular 
value  decomposition  for  all  input  and  output  powers  {ni.rio)  6  [0,  5]^  C 
the  calculation  of  the  maximum  relative  input  and  of  the  maximum  output 
errors  as  defined  by  eqs.  (48)  and  (49)  with  fn  ~  Ifb  leads  to  the  results 


1037 


Elimination  of  row  number... 


Frequency  [Hz] 


Frequency  [Hz] 

Figure  2:  Frequency  dependent  condition  numbers 

depicted  in  Fig.  3.  For  a  model  realisation  with  [ni.rio)  =  (0,2)  the  values 
of  the  maximum  relative  errors  are  approximately  ej  7.1  •  and  eo  ~ 
9.0-10“”^  which  corresponds  to  the  computational  accuracy.  Using  this  model 
the  estimates  of  the  filter  matrices  correspond  within  the  computational 
accuracy  to  those  of  the  ‘true’  foundation  model. 

Using  variations  of  the  force  fpB  between  the  bounds  /max,  /min  froni 
the  second  criterion  the  associated  upper  and  lower  bounds  of  the  relative 
input  and  output  error  have  been  calculated.  The  influence  of  the  variation 


1038 


Firgure  3:  Maximum  relative  input  and  output  error  for  different 
input  powers  rii  and  output  powers  rio  using  the  force  vector  fpB 


of  the  bearing  stiffness  of  50%  leads  via  the  associated  variation  of  the 
force  fpB  to  drastic  variations  of  the  relative  input  and  output  errors.  The 
difference  of  upper  and  lower  bound  of  the  maximum  relative  input  error  is 


1039 


Output  power 


Figure  4:  Maximum  relative  input  and  output  error  for  different 
input  powers  rii  and  output  powers  Uo  using  the  optimised  force 

vector  Jh 

of  order  100  and  that  of  the  maximum  relative  output  error  is  approximately 
LOl 


The  situation  is  different  using  the  optimised  force  fn  in  order  to  es¬ 
timate  the  filter  parameters  of  the  model  H.  For  each  input  and  output 
power  {ni,no)  e  [0,25]^  C  the  maximum  relative  input  error  and  the 
maximum  relative  output  error  have  been  calculated.  The  result  is  shown  in 
Fig.  4.  For  a  maximum  relative  input  error  e;  6.5  •  10“®  a  filter  model  of 
degree  12  is  available  with  the  powers  (n^,  =  (4, 8).  This  model  produces 

a  maximum  relative  output  error  eo  ^  10“®. 

Analogous  to  the  robustness  investigations  for  fpB  now  for  the  force 
/h  the  upper  and  lower  bounds  of  the  relative  input  and  of  the  relative 
output  error  due  to  the  random  variation  in  the  bearing  models  have  been 
calculated.  For  the  chosen  model  with  powers  (nj,no)  =  (4, 8)  the  difference 
of  upper  and  lower  bound  of  the  relative  input  error  as  well  that  of  the 
relative  output  error  are  of  about  the  same  order  10“^.  Thus,  compared 
with  the  order  of  variation  100  and  10"^  of  the  direct  foundation  model 
estimate  the  estimate  of  H  is  robust  with  respect  to  changes  in  the  bearing 
model. 


5  CONCLUSION 

In  this  paper  a  method  is  introduced  which  enables  the  decoupling  of  the 
two  problems  of  model  estimation  and  of  sensitivity  of  the  foundation  model 
with  respect  to  the  model  of  the  bearings.  The  method  produces  an  opti¬ 
mised  choice  of  the  input /output  equation  which  provides  a  transfer  func¬ 
tion  estimation  that  is  robust  with  respect  to  deviations  in  the  model  of 
the  bearings.  For  the  foundation  model  estimation  a  filter  model  is  intro¬ 
duced.  This  modelling  strategy  has  the  advantage  of  leading  to  a  linear 
inverse  problem.  The  disadvantage  is  that  with  increasing  model  degree 
the  equation  error  can  be  made  arbitrarily  small.  Because  this  error  should 
not  become  smaller  than  the  accuracy  of  the  data,  a  cut-off  limit  has  to  be 
determined  a  priori.  Further  investigations  should  allow  the  cut-off  limit  to 
be  related  to  the  data  errors. 

REFERENCES 

1.  Feng,  N.S.  and  Hahn,  E.J.,  Including  Foundation  Effects  on  the  Vi¬ 
bration  Behaviour  of  Rotating  Machinery.  Mechanical  Systems  and 
Signal  Processing,  1995  Vol.  9,  No.  3,  pp.  243-256. 

2.  Friswell,  M.I.,  Lees,  A.W.  and  Smart,  M.G.,  Model  Updating  Tech¬ 
niques  Applied  to  Turbo-Generators  Mounted  on  Flexible  Founda¬ 
tions.  NAFEMS  Second  International  Conference:  Structural  Dynam¬ 
ics  Modelling  Test,  Analysis  and  Correlation,  Glasgow:  NAFEMS, 
1996  pp.  461-472. 


1041 


3.  Lees,  A.W.  and  Friswell,  M.L,  Estimation  of  Forces  Exerted  on  Ma¬ 
chine  Foundations.  Identification  in  Engineering  Systems,  Wiltshire; 
The  Cromwell  Press  Ltd.,  1996,  pp.  793-803. 

4.  Smart,  M.G,,  Friswell,  M.L,  Lees,  A.W.  and  Prells,  U.,  Errors  in 
estimating  turbo-generator  foundation  parameters.  In  Proceedings 
ISMA21  -  Noise  and  Vibration  Engineering,  ed.  P.  Sas,  Katholieke 
Universiteit  Leuven,  Belgium,  1996,  Vol.  II,  pp.  1225-1235 

5.  Zanetta,  G.A.,  Identification  Methods  in  the  Dynamics  of  Turbogener¬ 
ator  Rotors.  The  International  Conference  on  Vibrations  in  Rotating 
Machinery,  IMechE,  C432/092,  1992,  pp.  173-181. 

6.  Friswell,  M.L  and  Mottershead,  J.E.,  Finite  Element  Model  Updating 
in  Structural  Dynamics.  Dordrecht,  Boston,  London:  Kluwer  Aca¬ 
demic  Publishers,  1995. 

7.  Natke,  H.G.,  Einfuhrung  in  die  Theorie  und  Praxis  der  Zeitreihen- 
und  Modalanalyse  -  Identifikation  schwingungsfdhiger  elastomechanis- 
cher  Systeme.  Braunschweig,  Wiesbaden:  Friedrich  Vieweg  Sz  Sohn, 
1993. 

8.  Natke,  H.G.,  Lallement,  G.,  Cottin,  N.  and  Prells,  U.,  Properties  of 
Various  Residuals  within  Updating  of  Mathematical  Models.  Inverse 
Problems  in  Engineering,  Vol.  1,  1995,  pp.  329-348. 

9.  Natke,  H.G.,  What  is  a  true  mathematical  model?  -  A  discussion  of 
system  and  model  definitions.  Inverse  Problems  in  Engineering,  1995, 
Vol.  1,  pp.  267-272. 

10.  Gawronski,  W.  and  Natke,  H.G.,  On  ARMA  Models  for  Vibrating 
Systems.  Probabilistic  Engineering  Mechanics,  1986,  Vol.  1,  No.  3, 
pp.  150-156. 

11.  Baumeister,J.,  Stable  Solution  of  Inverse  Problems.  Braunschweig, 
Wiesbaden:  Friedrich  Vieweg  &;  Sohn,  1987. 

ACKNOWLEDGEMENTS 

The  authors  are  indebted  to  Nuclear  Electric  Ltd  and  Magnox  Pic  for  fund¬ 
ing  the  research  project  BB/G/40068/A  to  develop  methods  which  enables 
the  estimation  of  the  influence  of  the  foundation  on  the  dynamics  of  the 
rotor.  Dr.  Friswell  gratefully  acknowledges  the  support  of  the  Engineering 
and  Physical  Sciences  Research  Council  through  the  award  of  an  Advanced 
Fellowship. 


1042 


MATHEMATICAL  HYSTERESIS  MODELS  AND 
THEIR  APPLICATION  TO  NONLINEAR 
ISOLATION  SYSTEMS 


Y.Q.  Ni,  J.M.  Ko  and  C.W.  Wong 

Department  of  Civil  and  Structural  Engineering 
The  Hong  Kong  Polytechnic  University,  Hong,  Kong 

Abstract 

Two  mathematical  hysteresis  models,  the  Duhem-Madelung  (DM) 
model  and  the  Preisach  model,  are  introduced  to  represent  the 
hysteretic  behavior  inherent  in  nonlinear  damping  devices.  The  DM 
model  generates  the  hysteresis  with  local  memory.  Making  use  of  the 
Duhem  operator,  the  constitutive  relation  can  be  described  by  single¬ 
valued  functions  with  two  variables  in  transformed  state  variable 
spaces.  This  makes  it  feasible  to  apply  the  force-surface  nonparametric 
identification  technique  to  hysteretic  systems.  The  Preisach  model  can 
represent  the  hysteresis  with  nonlocal  memory.  It  is  particularly 
suitable  for  describing  the  selective-memory  hysteresis  which  appears 
in  some  friction-type  isolators.  An  accurate  frequency-domain  method 
is  developed  for  analyzing  the  periodic  forced  vibration  of  hysteretic 
isolation  systems  defined  by  these  models.  A  case  study  of  wire-cable 
vibration  isolator  is  illustrated. 


1 .  Introduction 

The  dynamic  response  of  a  structure  is  highly  dependent  on 
the  ability  of  its  members  and  connections  to  dissipate  energy  by 
means  of  hysteretic  behavior.  The  assessment  of  this  behavior 
can  be  done  by  means  of  experimental  tests  and  the  use  of 
analytical  models  that  take  into  account  the  main  characteristics 
of  this  nonlinear  mechanism.  Although  a  variety  of  hysteresis 
models  have  been  proposed  in  the  past  decades,  many  structural 
systems  exhibit  more  complicated  hysteretic  performance  (mainly 
due  to  stiffness  or/ and  strength  degrading)  which  the  models  in 
existence  are  reluctant  and  even  inapplicable  to  depict  [1,2].  On 
the  other  hand,  nonlinear  vibration  isolation  has  recently  been 
recognized  as  one  of  effective  vibration  control  techniques.  In 


1043 


particular,  hysteretic  isolation  devices  have  got  wide  applications 
owing  to  their  good  diy  friction  damping  performance.  These 
hysteretic  isolators  may  exhibit  very  complicated  features  such 
as  asymmetric  hysteresis,  soft-hardening  hysteresis,  nonlocal 
selective-memory  hysteresis  [3-5].  None  of  the  models  available 
currently  in  structural  and  mechanical  areas  can  represent  all 
these  hysteresis  characteristics.  Other  more  elaborate  hysteresis 
models  need  to  be  established  for  this  purpose. 

In  reality,  hysteresis  phenomenon  occurs  in  many  different 
areas  of  science,  and  has  been  attracting  the  attention  of  many 
investigators  for  a  long  time.  However,  the  true  meaning  of 
hysteresis  varies  from  one  area  to  another  due  to  lack  of  a 
stringent  mathematical  definition  of  hysteresis.  Fortunately, 
because  of  the  applicative  interest  and  obvious  importance  of 
hysteresis  phenomenon,  Russian  mathematicians  in  1970’s  and 
the  Western  mathematicians  in  1980’s,  began  to  study  hysteresis 
systematically  as  a  new  field  of  mathematical  research  [6,7].  They 
also  deal  with  the  hysteresis  models  proposed  by  physicists  and 
engineers  in  various  areas,  but  they  separate  these  models  from 
their  physical  meanings  and  formulate  them  in  a  purely 
mathematical  form  by  introducing  the  concept  of  hysteresis 
operators.  Such  mathematical  exposition  and  treatment  can 
generalize  a  specific  model  from  a  particular  area  as  a  general 
mathematical  model  which  is  applicable  to  the  description  of 
hysteresis  in  other  areas.  In  this  paper,  two  mathematical 
hysteresis  models  are  introduced  and  the  related  problems  such 
as  identification  and  response  analysis  encountered  in  their 
application  to  nonlinear  isolation  systems  are  addressed. 

2.  Definition  of  Hysteresis 

Hysteresis  loops  give  the  most  direct  indication  of  hysteresis 
phenomena.  But  it  is  intended  here  to  introduce  a  mathematical 
definition  of  hysteresis.  Let  us  consider  a  constitutive  law:  u  r, 
which  relates  an  input  variable  u(t)  and  an  output  variable  r\t). 
For  a  structural  or  mechanical  system,  u(t)  denotes  displacement 
(strain);  r{t)  represents  restoring  force  (stress);  t  is  time.  We  can 
define  hysteresis  as  a  special  type  of  memoiy-based  relation 


1044 


between  u(t)  and  r(t).  It  appears  when  the  output  r[t)  is  not 
uniquely  determined  by  the  input  u(t)  at  the  same  instant  t,  but 
instead  r(t)  depends  on  the  evolution  of  u  in  the  interval  [0,  t]  and 
possibly  also  on  the  initial  value  ro,  i.e. 

r(t)  =  iR[u(-),ro](t)  (1) 

where  the  memory-based  functional  iR[u(-),ro](t)  is  referred  to  as 
hysteresis  operator.  In  order  to  exclude  viscosity-type  memory 
such  as  those  represented  by  time  convolution,  we  require  that  91 
is  rate-independent,  i.e.  that  r(t)  depends  just  on  the  range  of  u  in 
[0,  t]  and  on  the  order  in  which  values  have  been  attained,  not  on 
its  velocity.  In  reality,  memory  effects  may  be  not  purely  rate 
independent  as  hysteresis  is  coupled  with  viscosity- type  effects. 
However,  as  shown  later,  in  most  cases  the  rate  independent 
feature  of  hysteresis  is  consistent  with  experimental  findings, 
especially  when  evolution  (variation  in  time)  is  not  too  fast. 

3.  Duhem-Madelung  (DM)  Model 

3.1  Formulation 

The  DM  model  can  be  defined  with  or  without  referring  to  a 
confined  hysteresis  region.  For  the  structural  or  mechanical 
hysteretic  systems,  it  is  not  necessary  to  introduce  the  notion  of 
bounded  curves  because  there  exists  neither  the  saturation  state 
nor  the  major  loop.  In  this  instance,  the  DM  model  establishes  a 
mapping  (named  Duhem  operator)  91:  (u,  ro)  rby  postulating  the 


following  Cauchy  problem  [7,8] 

=  g,  (li. '■)  ■  w,  (t)  -  02 (u, r)  •  u_ (t)  (2a) 

r(0)  =  r„  (2b) 

where  an  overdot  denotes  the  derivative  with  respect  to  t;  gi{u,r] 
and  g2{ii,r)  are  referred  to  as  ascending  and  descending  functions 
(curves)  respectively;  and 

tijt)  =  max[0,u(t)]  =  t  [|u(t)|  +  u(t)]  (3a) 

u.(t)  =  min[0,u(t)]  =  ^[|u(t)|  -  u(f)]  (3b) 

Eq.  (2a)  can  be  rewritten  as 

f(f)  =  p[u,r,sgn(u)]-u(t)  (4) 


1045 


u>  0 
u<0 


(5) 


in  which  the  describing  function  has  the  form 


g{u,r,sgn{u)] 


fg{u,r,l)  =  g,{u,r) 
[g[u,r-l)  =  g^(u,r) 


It  is  obvious  that  the  DM  model  is  rate  independent.  In 
addition,  it  is  specially  noted  that  in  this  constitutive  law  the 
output  r{t)  is  not  directly  dependent  on  the  entire  history  of  u{t) 
through  [0,t];  but  instead  depends  only  on  the  local  histoiy 
covered  since  the  last  change  of  sgn(u)  and  on  the  value  of  the 
output  at  this  switching  instant.  It  means  that  the  output  can 
only  change  its  character  when  the  input  changes  direction.  As  a 
consequence,  the  DM  model  usually  represents  the  hysteresis 
with  local  memory  except  that  the  functions  gi(u,?)  and  g2{u,7)  are 
re-specified  as  hysteresis  operators. 

Within  the  framework  of  DM  formulation,  the  ascending  and 
descending  functions  gi(u,T]  and  g2(u,r)  are  just  required  to  fulfill 
suitable  regularity  conditions  and  need  not  to  be  specified  in 
specific  expressions,  so  both  the  form  and  parameters  of  the 
functions  can  be  fine-tuned  to  match  experimental  findings.  On 
the  other  hand,  the  DM  formulation  can  deduce  a  wide  kind  of 
differential-type  hysteresis  models  such  as  Bouc-Wen  model, 
Ozdemir’s  model,  Yar-Hammond  bilinear  model  and  Dahfs 


frictional  model.  For  the  Bouc-Wen  model 

r(t)  =  K.u{t)  +  z(t)  (6a) 

z{t)  =  au(t]  -  P|ti(f)|z(f)|z(t)|'”‘  -  YU(t)|z(f )|"  (6b) 

it  corresponds  to  the  DM  model  with  the  specific  ascending  and 
descending  functions  as 

grj(u,r)  =  a  +  K  “  [y  +  p  sgn(r  -Ku)]|r  -Kup  (7a) 

g2(w,r)  =  a  +  K  -  [y  -  p  sgn(r  -Kw)]|r  -ku|”  (7b) 

and  for  the  Yar-Hammond  bilinear  model 

f{t)  =  {a  - y  sgn(ii)  sgn[r  -  p  sgn(ii)]}ii  (8) 

its  describing  function  is  independent  of  u(t)  as  follows 

5f[u,r,sgn(u)l  =  gf[r,sgn(u)l  =  a  -y  sgn(u)sgn[r  -  p  sgn(u)]  (9) 


Hence,  the  Duhem  operator  also  provides  an  accessible  way 
to  construct  novel  hysteresis  models  by  prescribing  specific 


1046 


expressions  of  the  ascending  and  descending  functions. 
Following  this  approach,  it  is  possible  to  formulate  some  models 
which  allow  the  description  of  special  hysteretic  characteristics 
observed  in  experiments,  such  as  soft-hardening  hysteresis, 
hardening  hysteresis  with  overlapping  loading  envelope,  and 
asymmetric  hysteresis  [5]. 

3.2  Identification 

System  identification  techniques  are  classified  as  parametric 
and  nonparametric  procedures.  The  parametric  identification 
requires  that  the  structure  of  system  model  is  a  priori  known.  The 
advantage  of  nonparametric  identification  methods  is  that  they 
do  not  require  a  priori  the  knowledge  of  system  model.  The  most 
used  nonparametric  procedure  for  nonlinear  systems  is  the  force 
mapping  (or  called  force  surface)  method  [9].  This  method  is 
based  on  the  use  of  polynomial  approximation  of  nonlinear 
restoring  force  in  terms  of  two  variables — ^the  displacement  u(t) 
and  the  velocity  u{t) .  For  nonlinear  hysteretic  systems,  however, 
the  hysteretic  restoring  force  appears  as  a  multivalued  function 
with  respect  to  the  variables  u{t)  and  u(t)  due  to  its  history- 
dependent  and  non-holonomic  nature.  This  renders  the  force 
mapping  method  inapplicable  to  hysteretic  systems,  although 
some  efforts  have  been  made  to  reduce  the  multivaluedness  of 
the  force  surface  [10,11]. 

One  of  the  appealing  virtues  of  the  DM  model  is  that  it  can 
circumvent  this  difficulty.  Making  use  of  the  Duhem  operator, 
the  hysteretic  constitutive  relation  of  Eq.(l)  is  described  by  two 
continuous,  single-valued  functions  gi(u,r)  and  g2{u,r)  in  terms  of 
the  displacement  u(t)  and  the  restoring  force  r{t).  Thus,  single¬ 
valued  “force”  surfaces  gi{u,r)  and  g2{u,7)  can  be  formulated  in 
the  subspaces  of  the  state  variables  {u,r,gi)  and  (u,r,g2),  and  can 
be  identified  by  using  the  force  mapping  technique.  Following 
this  formulation,  a  nonparametric  identification  method  is 
developed  by  the  authors  [12].  In  this  method,  the  functions 
g\(u,7)  and  g2{u,7)  are  expressed  in  terms  of  shifted  generalized 
orthogonal  polynomials  with  respect  to  u  and  r  as  follows 

gM,r)  =  i  =  0’'(u)G<''0(r)  (10a) 

i=lj=l 


1047 


(10b) 


g^r]  =  z  i(|>,.(r)gf(i>,(u)  =  0^(u)G'^'<l>(r) 

I=lj=l 

where  =  [gf  ]^xn  =[5'zf]mxn  ^re  called  the  expansion- 

coefficient  matrices  of  gi{u,r)  and  g2{u,T].  Some  algorithms  have 
been  proposed  to  estimate  the  values  of  these  coefficient  matrices 
based  on  experimentally  observed  input  and  output  data.  It 
should  be  noted  that  here  the  vectors  0(u)  and  <^{r)  are  shifted 
generalized  orthogonal  polynomials  [13].  They  are  formulated  on 
the  basis  of  common  recurrence  relations  and  orthogonal  rule, 
and  cover  all  kinds  of  individual  orthogonal  polynomials  as  well 
as  non- orthogonal  Taylor  series.  Consequently,  they  can  obtain 
specific  polynomial-approximation  solutions  of  the  same. problem 
in  terms  of  Chebyshev,  Legendre,  Laguerre,  Jacobi,  Hermite  and 
Ultraspherical  polynomials  and  Taylor  series  as  special  cases. 

4.  Preisach  Model 


4. 1  Formulation 

The  intent  of  introducing  the  Preisach  model  is  to  supply  the 
lack  of  a  suitable  hysteresis  model  in  structural  and  mechanical 
areas,  which  is  both  capable  of  representing  nonlocal  hysteresis 
and  mathematically  tractable.  Experiments  revealed  that  the 
hysteretic  restoring  force  of  some  cable-type  vibration  isolators 
relates  mainly  to  the  peak  displacements  incurred  by  them  in  the 
past  deformation  [3].  It  will  be  shown  that  the  Preisach  model  is 
especially  effective  in  representing  such  nonlocal  but  selective- 
memory  hysteresis,  in  which  only  some  past  input  extrema  (not 
the  entire  input  variations)  leave  their  marks  upon  future  states 
of  hysteresis  nonlinearities. 

The  Preisach  model  is  constructed  as  a  superposition  of  a 
continuous  family  of  elementary  rectangular  loops,  called  relay 
hysteresis  operators  as  shown  in  Fig.  1.  That  is  [7,14], 

r(f)  = 'R[u(-)](f)  =  j||^(a,P)y„D[u(t)]dadf5  (11) 

a>P 

where  ia(a,|3)  >  0  is  a  weight  function,  usually  with  support  on  a 
bounded  set  in  the  (a,p)-plane,  named  Preisach  plane;  Ya.p[^(^)]  is 
the  relay  hysteresis  operator  with  thresholds  a  >  p.  Outputs  of 


1048 


these  simplest  hysteresis  operators  have  only  two  values  +1  and 
-1,  so  can  be  interpreted  as  two-position  relays  with  “up”  and 
“down”  positions  corresponding  to  ya,p[^^(^)l=+l  and 


1 

+1 

p 

a 

-1 

Fig.  1  Relay  Hysteresis  Operator 


Hence,  the  Preisach  model  of  Eq.(ll)  can  be  interpreted  as  a 
spectral  decomposition  of  the  complicated  hysteresis  operator  iR, 
that  usually  has  nonlocal  memory,  into  the  simplest  hysteresis 
operators  7ct,p  with  local  memory.  In  the  following,  we  illustrate 
how  the  model  depicts  the  nonlocal  selective-memory  feature. 

Consider  a  triangle  T  in  the  half-plane  a  >  (3  as  shown  in 
Fig.2.  It  is  assumed  that  the  weight  function  )i(a,p)  is  confined  in 
the  triangle  T,  i.e.  }i(a,p)  is  equal  to  zero  outside  T.  Following  the 
Preisach  formulation,  at  any  time  instant  t,  the  triangle  T  can  be 
subdivided  into  two  sets:  S^(t)  consisting  of  points  (a,P)  for  which 
the  corresponding  Yc^.p-operators  are  in  the  “up”  position;  and  S^(t) 
consisting  of  points  {a,P)  for  which  the  corresponding  Ya.p“ 
operators  are  in  the  “down”  position.  The  interface  L(t)  between 
S^(t)  and  Sr(t)  is  a  staircase  line  whose  vertices  have  a  and  p 
coordinates  coinciding  respectively  with  local  maxima  and 
minima  of  input  at  previous  instants  of  time.  The  nonlocal 
selective-memory  is  stored  in  this  way.  Thus,  the  output  r(^)  a-t 
any  instant  t  can  be  expressed  equivalently  as  [14] 

r{t)=  J|^(a,P)dadp-  |jM,(a,P)dadp  (12) 

S'(£)  S'(t) 

It  should  be  noted  that  the  Preisach  model  does  not 
accumulate  all  past  extremum  values  of  input.  Some  of  them  can 
be  wiped  out  by  sequent  input  variations  following  the  wiping-out 
property  {deletion  rule):  each  local  input  maximum  wipes  out  the 


1049 


vertices  of  L(t)  whose  a-coordinates  are  below  this  maximum,  and 
each  local  minimum  wipes  out  the  vertices  whose  p-coordinates 
are  above  this  minimum.  In  other  words,  only  the  alternating 
series  of  dominant  input  extrema  are  stored  by  the  Preisach 
model;  all  other  input  extrema  are  erased. 


Fig.  2  Input  Sequence  and  Preisach  Plane 


4.2  Identification 


It  is  seen  from  Eq.(l  1)  that  the  Preisach  model  is  governed  by 
the  weight  function  |i(a,p)  after  determining  L(t)  which  depends 
on  the  input  sequence.  !i(a,P)  is  a  single-valued  function  with 
respect  to  two  variables  a  and  p.  Hence,  the  aforementioned 
nonparametric  identification  method  can  be  also  implemented  to 
identify  iLL(a,p)  by  expanding  it  in  a  similar  expression  to  Eq.(lO). 
An  alternating  approach  is  to  define  the  following  function 


H'(a',P')  =  |j|a(a,p)dadp  =  ^.[^  ^(ot,P)da]dp  (13) 


r(cc'.3') 


where  T(a',p')  is  the  triangle  formed  by  the  intersection  of  the 
line  a  =  a' ,  p  =  p'  and  a  =  p .  Differentiating  Eq.(13)  yields 


^(a',P')  =  -- 


aa'ap' 


(14) 


Thus,  the  force  mapping  identification  technique  can  be  applied 
to  determine  H(a,p)  consistent  with  the  experimental  data,  and 
then  ^(a,p)  is  obtained  by  Eq.(14). 


5.  Steady-State  Response  Analysis 

Hysteretic  systems  are  strongly  nonlinear.  A  study  of  the 
steady- state  oscillation  is  one  of  the  classical  problems  of 


1050 


nonlinear  systems.  Usually,  the  dynamic  behavior  of  a  nonlinear 
system  is  represented  by  its  resonant  frequency  and  frequency 
response  characteristics.  In  the  following,  an  accurate  frequency- 
domain  method  accommodating  multiple  harmonics  is  developed 
to  analyze  the  periodically  forced  response  of  hysteretic  systems 
defined  by  mathematical  hysteresis  models. 

Fig. 3  shows  a  single-degree-of- freedom  hysteretic  oscillator 
with  mass  m,  viscous  damping  coefficient  c,  and  linear  stiffness 
ky  subjected  to  an  external  excitation  F(t),  for  which  the  governing 
equation  of  motion  is 

m  •  u(t)  +  c  •  u(t)  +  k  '  u[t)  +  r{t)  =  F(t)  (15) 

where  the  hysteretic  restoring  force  r(t)  is  represented  by  the  DM 
model  as  Eq.(4).  It  is  worth  noting  that  for  the  kinetic  equation 
Eq.(15),  the  excitation  is  F(t)  and  the  response  is  u(t);  and  for  the 
hysteretic  constitutive  law  Eq.(4),  u(t)  is  input  and  r{t)  is  output. 
The  causal  relationship  is  different. 


Fig.  3  Single -Degree-of-Freedom  Hysteretic  System 

Due  to  the  hereditary  nature  of  the  hysteresis  model,  it  is 
difficult  to  directly  solve  the  kinetic  equation  Eq.(15)  by  iteration. 
Here,  Eq.(15)  is  only  used  to  establish  the  relation  between  the 
harmonic  components  of  u(t)  and  r(t).  Suppose  that  the  system  is 
subjected  to  a  general  periodic  excitation  F[t)  with  known 
harmonic  components  F={Fq  F^  F2  •••  F2  .  The  multi¬ 
harmonic  steady-state  response  can  be  expressed  as 

a  ^  ^  . 

u{t)  =  ~ +  a j  cos  j(Dt +  a j  sin  jcot  (16) 

2  j=i 


1051 


in  which  a={ao  a2  •••  a^v  ^2  unknown 

vector  containing  the  harmonic  components  of  u(t).  Introducing 

Eq.(16)  into  Eq.(15)  and  using  the  Galerkin  method  provide 

ro^F^-k-a^  (17a) 

Tj  =  Fj  -c-citj-a]  [j  =  1,2,  •••  ,  N)  (17b) 

Tj  =Fj  +c-coj-a^.  -(/c-m-coV^)-a*  (j  =  1,  2,  ,N)  (17c) 

where  r={ro  q  •••  r^}'^  is  the  harmonic  vector  of 

the  hysteretic  restoring  force  r(t).  Referring  to  the  hysteretic 
constitutive  law,  we  define  the  determining  equation  as 

D(t)  =  r{t)  -  g[u,r,sgn(u)]  •  u{t)  (18) 

When  a  is  the  solution  of  u(t),  applying  the  Galerkin  method 
into  Eq.(18)  and  considering  Eq.(17)  achieve 

d{a)  =  0  (19) 

where  the  vector  d{a)={dQ  ^3  --dj^  d^  d^  "•  is  comprised 

of  the  harmonic  components  of  D(t)  corresponding  to  a.  An 

efficient  procedure  to  seek  the  solution  of  Eq.(19)  is  the 
Levenberg-Marquardt  algorithm  with  the  iteration  formula 

where  the  Jacobian  matrix  J[a(^)]  =  dd(a) / da\a=a^^)  ;  9^  is  the 

Levenberg-Marquardt  parameter  and  I  is  identity  matrix. 

At  each  iteration,  the  function  vector  and  Jacobian 

matrix  should  be  recalculated  with  updated  values  of 

Here,  a  frequency/ time  domain  alternation  scheme  by  FFT  is 
introduced  to  evaluate  the  values  of  d(a)  and  J[a)  at  d{a) 

and  dd[a)/da  are  known  to  be  the  Fourier  expansion  coefficients 
of  D(t)  and  dD(t)/ da  respectively.  For  a  given  a(^)  and  known  F,  the 
corresponding  r[a(^)]  is  obtained  from  Eq.(17),  and  the  inverse 
FFT  is  implemented  for  and  r[a(^)]  to  obtain  all  the  time 
domain  discrete  values  of  u{t] ,  u(t) ,  r(t)  and  f(t)  over  an  integral 
period.  Then  the  time  domain  discrete  values  of  the  function 
D{u,u,r,r,t) ,  corresponding  to  a=a(^),  are  evaluated  from  Eq.(18). 
Making  use  of  forward  FFT  to  these  time  domain  discrete  values 
of  D{u,u,r,r,t) ,  the  values  of  function  vector  d[a(^)]  are  obtained. 


1052 


Similarly,  the  partial  differential  dD{t)  /  da  can  be  analytically 
evaluated  in  the  time  domain.  Forward  FFT  to  the  time  domain 
values  of  dD(t)  /  da  at  ct=a(^)  gives  rise  to  dd[ai^)]/da. 

6.  Case  Study 

Wire-cable  vibration  isolators  are  typical  hysteretic  damping 
devices.  Dynamic  tests  show  that  their  hysteresis  behaviors  are 
almost  independent  of  the  frequency  in  the  tested  frequency 
range  [4,15].  Experimental  study  and  parametric  modelling  of  a 
wire-cable  isolator  have  been  carried  out  [5].  Fig. 4  shows  the 
experimental  hysteresis  loops  in  shear  mode.  It  is  seen  that  for 
relatively  small  deformations,  the  isolator  exhibits  softening 
hysteresis  loops.  When  large  displacements  are  imposed,  the 
stiffness  of  the  loops  becomes  smoothly  hard.  This  nature  is 
referred  to  as  soft-hardening  hysteresis.  Based  on  the  Bouc-Wen 
model,  a  parametric  identification  was  performed  to  model  these 
hysteresis  loops,  but  the  result  is  unsatisfactory.  This  is  due  to 
the  fact  that  the  Bouc-Wen  model  cannot  represent  such  soft- 
hardening  nature  of  hysteresis. 


Fig.  4  Experimental  Hysteresis  Loops 


We  now  use  the  DM  model  to  represent  these  hysteresis 
loops,  and  perform  a  nonparametric  identification  to  determine 
the  functions  gi(u,7)  and  g2(u,7).  The  simplest  Taylor  series  are 
adopted,  i.e.  (i),(r)  =  (r  /  and  (|)j(u)  =  (u  j  UqY~^  (ro  =  20.0  and 
uo  =  2.0).  Fig. 5  shows  the  identified  “force”  surfaces  of  gi{u,T)  and 
g2(u,r)  by  taking  m=n~S.  Fig. 6  presents  the  theoretical  hysteresis 


1053 


loops  generated  by  the  DM  model  using  the  identified  g\(u,T)  and 
g2(u,}).  It  is  seen  that  the  modeled  hysteresis  loops  are  agreeable 
to  the  observed  loops.  In  particular,  the  soft-hardening  nature  is 
reflected  in  the  modeled  hysteresis  loops. 


-200 


Fig.  6  Modeled  Hysteresis  Loops  by  DM  Model 


After  performing  the  modelling  of  hysteretic  behavior,  the 
dynamic  responses  of  hysteretic  systems  can  be  predicted  by  the 
developed  method.  Fig. 7  shows  a  vibration  isolation  system 
installed  with  wire-cable  isolators  in  shear  mode.  It  is  subjected 
to  harmonic  ground  acceleration  excitation  Xg[t)=Acos2Kft.  The 

equation  of  motion  of  the  system  is  expressed  as 

m-u(t)  + K  ■r(t)  = -rri' Xg(t)  (21) 

where  m  is  the  mass  of  the  system;  K  is  number  of  the  isolators 
installed.  u{t)  is  the  displacement  of  the  system  relative  to  the 
ground.  r(t)  is  the  restoring  force  of  each  isolator  and  has  been 
determined  from  nonparametric  identification. 

Fig. 8  illustrates  the  predicted  frequency-response  curves  of 
the  relative  displacement  when  7n=6kg  and  K=2.  The  excitation 
amplitude  A  is  taken  as  0.25g,  O.SOg,  0.35g,  0.40g  and  0.45g 
respectively.  The  frequency-response  curves  show  clearly  the 
nonlinear  nature  of  the  wire-cable  isolation  system. 


i 

I 


Xg(t) 


Fig.  7  Vibration  Isolation  System  with  Wire -Cable  Isolators 


Fig.  8  Frequency  Response  Curves  of  Relative  Displacement 


1055 


7.  Concluding  Remarks 


This  paper  reports  a  preliminaiy  work  of  introducing  the 
mathematical  hysteresis  models  in  structural  and  mechanical 
areas.  It  is  shown  that  a  wide  kind  of  differential  hysteresis 
models,  which  are  extensively  used  at  present,  can  be  derived 
from  the  Duhem-Madelung  (DM)  model.  Thus,  the  mathematical 
properties  concerning  the  DM  model  are  also  possessed  by  these 
models.  Two  potential  advantages  appear  when  the  DM 
formulation  is  used.  Firstly,  it  allows  to  apply  the  force  mapping 
technique  to  hysteretic  systems.  Secondly,  it  provides  an 
approach  to  construct  novel  differential  models  which  reflect 
some  special  hysteretic  characteristics.  The  Preisach  model  is 
shown  to  be  capable  of  representing  nonlocal  hysteresis  and 
mathematically  tractable.  It  offers  a  more  accurate  description  of 
several  observed  hysteretic  phenomena.  Emphasis  is  placed  on 
demonstrating  the  selective-memory  nature  of  this  hysteresis 
model.  The  case  study  based  on  experimental  data  of  a  wire- 
cable  isolator  has  shown  the  applicability  of  the  mathematical 
hysteresis  model,  and  the  validity  of  the  steady-state  response 
analysis  method  proposed  in  the  present  paper. 

Acknowledgment:  This  study  was  supported  in  part  by  the 
Hong  Kong  Research  Grants  Council  (RGC)  and  partly  by  The 
Hong  Kong  Polytechnic  University.  These  supports  are  gratefully 
acknowledged. 


References 

1.  Azevedo,  J.  and  Calado,  L.,  ‘‘Hysteretic  behaviour  of  steel 
members:  analytical  models  and  experimental  tests”,  J. 
Construct  Steel  Research,  1994,  29,  71-94. 

2.  Kayvani,  K.  and  Barzegar,  F.,  “Hysteretic  modelling  of  tubular 
members  and  offshore  platforms”,  Eng.  Struct,  1996,  18,  93- 
101. 

3.  Lo,  H.R.,  Hammond,  J.K.  and  Sainsbury,  M.G.,  “Nonlinear 
system  identification  and  modelling  with  application  to  an 
isolator  with  hysteresis”,  Proc.  6th  Int  modal  Anal  Conf., 
Kissimmee,  Florida,  1988,  Vol.II,  1453-1459. 


1056 


4.  Demetriades,  G.F.,  Constantinou,  M.C.  and  Reinhorn,  A.M., 
“Study  of  wire  rope  systems  for  seismic  protection  of 
equipment  in  buildings”,  Eng.  Struct,  1993,  15,  321-334. 

5.  Ni,  Y.Q.,  “Dynamic  response  and  system  identification  of 
nonlinear  hysteretic  systems”,  PhD  Dissertation,  The  Hong 
Kong  Polytechnic  University,  Hong,  Kong,  November  1996. 

6.  Krasnosefskii,  M.A.  and  Pokrovskii,  A.V.,  Systems  with 
Hysteresis,  translated  from  the  Russian  by  M.  Niezgodka, 
Springer-Verlag,  Berlin,  1989. 

7.  Visintin,  A.,  Differential  Models  of  Hysteresis,  Springer-Verlag, 
Berlin,  1994, 

8.  Macki,  J.W.,  Nistri,  P.  and  Zecca,  P.,  “Mathematical  models 
for  hysteresis”,  SIAM  Review,  1993,  35,  94-123. 

9.  Masri,  S.F.  and  Caughey,  T.K.,  “A  nonparametric 

identification  technique  for  nonlinear  dynamic  problems”,  J. 
Appl  Mech,  ASME,  1979,  46,  433-447. 

10.  Lo,  H.R.  and  Hammond,  J.K.,  “Nonlinear  system  identification 
using  the  surface  of  nonlinearity  form:  discussion  on 
parameter  estimation  and  some  related  problems”,  Proc.  3rd 
Int  Conf  Recent  Adv.  Struct.  Dyn.,  Southampton,  UK,  1988, 
339-348. 

11. Benedettini,  F.,  Capecchi,  D.  and  Vestroni,  F.,  “Identification 
of  hysteretic  oscillators  under  earthquake  loading  by 
nonparametric  models”,  J.  Eng.  Mech.,  ASCE,  1995,  121,  606- 
612. 

12.  Ni,  Y.Q.,  Ko,  J.M.  and  Wong,  C.W.,  “Modelling  and 
identification  of  nonlinear  hysteretic  vibration  isolators”. 
Accepted  to  SPJE’s  4th  Annual  Symposium  on  Smart  Structures 
and  Materials  :  Passive  Damping  and  Isolation,  3-6  March 
1997,  San  Diego,  USA. 

13. Ni,  Y.Q.,  Wong,  C.W.  and  Ko,  J.M.,  “The  generalized 
orthogonal  polynomial  (GOP)  method  for  the  stability  analysis 
of  periodic  systems”,  Proc.  Int.  Conf.  Comput.  Methods  Struct. 
Geotech.  Eng.,  Hong  Kong,  1994,  Vol.II,  464-469. 

14. Mayergoyz,  I.D.,  Mathematical  Models  of  Hysteresis,  Springer- 
Verlag,  New  York,  1991. 

15.  Ko,  J.M.,  Ni,  Y.Q.  and  Tian,  Q.L.,  “Hysteretic  behavior  and 
empirical  modeling  of  a  wire-cable  vibration  isolator”,  Int.  J. 
Anal.  Exp.  Modal  Anal,  1992,  7,  111-127. 


1057 


1058 


The  identification  of  turbogenerator  foundation  models 
from  run-down  data 


M  Smart,  M  I  Friswell,  A  W  Lees,  U  Prells 

Department  of  Mechanical  Engineering 
University  of  Wales  Swansea,  Swansea  SA2  8PP  UK 
email:  m.smart@swansea.ac.nk 


ABSTRACT 

The  trend  of  placing  turbines  in  modern  power  stations  on  flexi¬ 
ble  steel  foundations  means  that  the  foundations  exert  a  considerable 
influence  on  the  dynamics  of  the  system.  In  general,  the  complex¬ 
ity  of  the  foundations  means  that  models  are  not  available  a  priori, 
but  rather  need  to  be  identified.  One  way  of  doing  this  is  to  use 
the  measured  responses  of  the  foundation  at  the  bearings  to  the 
synchronous  excitation  obtained  when  the  rotor  is  run  down.  This 
paper  discusses  the  implementation  of  such  an  estimation  technique, 
based  on  an  accurate  model  of  the  rotor  and  state  of  unbalance,  and 
some  knowledge  of  the  dynamics  of  the  bearings.  The  effect  of  errors 
in  the  bearing  model  and  response  measurements  on  the  identified 
parameters  is  considered,  and  the  instrumental  variable  method  is 
suggested  as  one  means  of  correcting  them. 


1  INTRODUCTION 

The  cost  of  failure  of  a  typical  turbine  in  a  modern  power  station  is 
very  high,  and  therefore  development  of  condition  monitoring  techniques 
for  such  machines  is  an  active  area  of  research.  Condition  monitoring  relies 
on  measuring  machine  vibrations  and  using  them  to  locate  and  quantify 
faults,  which  obviously  requires  an  accurate  dynamic  model  of  the  ma¬ 
chine.  Although  the  dynamic  characteristics  of  rotors  are  generally  well 
understood,  the  foundations  on  which  they  rest  are  not.  Since  the  founda¬ 
tions  are  often  quite  flexible,  they  can  contribute  considerably  to  the  rotor’s 
dynamic  behaviour. 

Finite  element  modelling  has  been  attempted  but  the  complexity  of 
the  foundations,  and  the  fact  that  they  often  differ  substantially  from  the 
original  drawings  rendered  the  technique  generally  unsuccessful [1].  Exper¬ 
imental  modal  analysis  is  another  possible  solution,  but  this  requires  that 
the  rotor  be  removed  from  the  foundation,  and  that  all  casings  remain  in 
place,  which  is  not  practical  for  existing  power  plant.  However,  mainte¬ 
nance  procedures  require  that  rotors  are  run  down  at  regular  intervals  and 
this  procedure  provides  forcing  to  the  foundation  over  the  frequency  range 


1059 


of  operation.  By  measuring  the  response  at  the  bearing  pedestals  (which 
is  already  performed  for  condition  monitoring  purposes)  an  input-output 
relation  for  the  foundation  may  be  obtained. 

Lees  [2]  developed  a  least-squares  method  to  calculate  the  foundation 
parameters  by  assuming  that  an  accurate  model  exists  for  the  rotor,  that 
the  state  of  unbalance  is  known  from  balancing  runs,  and  that  the  dynamic 
stiffness  matrices  of  the  bearings  can  be  calculated.  Although  bearing  mod¬ 
els  are  not  in  fact  well  characterised,  Lees  and  Friswell[3]  showed  that  the 
parameter  estimates  are  only  sensitive  to  the  bearing  stiffnesses  over  limited 
frequency  ranges,  which  can  be  calculated. 

Feng  and  Hahn [4]  followed  a  similar  approach  but  added  extra  informa¬ 
tion  by  measuring  the  displacements  of  the  shaft.  Zanetta[5]  also  measured 
the  shaft  displacements  but  included  the  bearing  characteristics  as  param¬ 
eters  to  be  estimated.  Although  any  extra  information  is  desirable  in  a 
parameter  estimation  routine  the  equipment  necessary  to  measure  these 
quantities  only  exists  in  the  newer  power  stations,  and  it  was  desired  to 
make  the  method  applicable  to  older  plant  as  well.  In  the  analysis  presented 
here,  the  measured  data  consists  of  the  motion  of  the  bearing  pedestals  in 
the  horizontal  and  vertical  directions,  although  not  necessarily  in  both  di¬ 
rections  at  every  bearing. 

2  THEORY 

2.1  Force  estimation 

If  D  is  the  dynamic  stiffness  matrix  of  a  structure  defined  as 

D{u)  K  ^luC  -oj^M  (1) 

where  M,C,K  are  the  mass,  damping  and  stiffness  matrices  then 

Dx  =  f  (2) 

where  x  is  response  and  f  force.  Referring  to  figure  1  it  is  seen  that  the  rotor 
is  connected  to  the  foundation  via  the  bearings.  It  is  assumed  that  good 
models  exist  for  both  rotor  and  bearings,  and  that  the  state  of  unbalance 
is  known.  The  implications  of  these  assumptions  will  be  discussed  later. 

The  dynamic  stiffness  equation  for  the  whole  system  may  be  written  as 


The  subscripts  r  and  /  refer  to  the  rotor  and  foundation  degrees-of-freedom 
respectively,  u  refers  to  the  unbalance  forces  and  b  to  the  bearing  forces. 


1060 


ROTOR 


Figure  1:  Rotor-bearing  system 


There  is  a  negative  sign  before  the  bearing  forces  ft,  since  they  refer  to  the 
forces  acting  on  the  bearings.  The  foundation  d.o.f  are  those  where  the  re¬ 
sponses  are  measured,  in  other  words  no  internal  d.o.f  are  represented.  Df 
therefore  represents  a  reduced  dynamic  stiffness  matrix.  The  response  mea¬ 
surements  will  not  be  the  total  vibration  level  at  the  bearings  but  rather  the 
vibrations  at  once-per-revolution  and  it  is  assumed  that  no  dynamic  forces 
at  this  frequency  will  be  transmitted  to  the  foundation  via  the  substructure 
onto  which  it  is  fixed. 

The  equation  for  the  bearings  in  the  global  coordinate  system  is 


/  Dbrr  Dbrf  W  ^7-  ^  ^  fbT\ 
V  Dij,  Dill  )\^s)  \fbs) 


(4) 


This  assumes  that  the  bearings  behave  as  complex  springs,  in  other 
words  they  have  negligible  inertia  and  no  internal  d.o.f.  Substituting  (4) 
into  (3)  we  have 


/  Dt„  Dtrl  \  (  ^'■  \  =  ( 

V  A/r  A//  J  [  J  A  °  / 

where 

/  Dtrl  0  \  f  Dirr  Di^l  \\ 

[Dtlr  DtllJ-[[  0  DlJ  +  [Diir  Dill)) 


(5) 

(6) 


1061 


and  where  subscript  t  refers  to  the  total  model.  Solving  equation  (5)  for 
Xr  leads  to 

Xr  —  J^trr  ifur  ~~ 

and  solving  equation  (4)  for  ft /  yields 

fbf  —  ^fhr  ~  ~^brr^r  ~  ^brf^f  (S) 

All  quantities  in  equation  (8)  are  known  either  from  assumed  models  {Dr,  D^) 
or  experiment  {xf,  /^r)-  This  calculated  force  fbf  may  then  be  used  to¬ 
gether  with  the  measured  responses  to  estimate  the  foundation  parameters. 

2.2  Foundation  parameter  estimation 

Once  the  forces  have  been  estimated,  the  foundation  parameters  must 
be  derived.  The  dynamic  stiffness  equation  for  the  foundation  is 


DfXf  —  fbf  (9) 

Although  D/  is  a  reduced  stiffness  matrix  it  is  assumed  that  it  has  the 
form  of  equation  (1).  Therefore  equation  (9)  may  be  written  as 

W{u)v  =  fbf  (10) 

where  is  a  column  vector  formed  from  the  elements  of  M,  C  and  K  and 
W  is  a  matrix  formed  from  the  response  vector  which  depends  explicitly  on 
u.  Clearly  this  is  an  under-determined  set  of  equations,  but  by  taking  mea¬ 
surements  at  many  frequencies  it  may  be  made  over-determined,  and  thus 
solvable  in  a  least  squares  sense.  Since  the  magnitude  of  the  mass,  damp¬ 
ing  and  stiffness  elements  normally  differ  by  several  orders-of-magnitude,  it 
was  found  expedient  to  scale  the  mass  parameters  by  o;^,  and  the  damping 
parameters  by  uJ,  where  u  is  the  mean  value  of  the  frequency. 

2.3  Errors  in  estimates 

It  is  necessary  now  to  examine  the  effect  of  errors  on  the  parameter  esti¬ 
mates.  Equation  (10)  is  of  the  form  Ax  =  b,  where  A  has  dimension  mxn. 
In  this  particular  case,  A  depends  on  the  measured  response  Xf,  whilst  b 
depends  on  the  measured  response,  applied  unbalance,  and  assumed  rotor 
and  bearing  models.  Therefore  the  estimated  parameters  will  be  sensitive 
to  the  following  errors: 

1.  Errors  in  the  rotor  model 

2.  Errors  in  the  bearing  model 

3.  Errors  in  the  state  of  unbalance 


1062 


4.  Errors  in  the  measured  foundation  response 

The  rotor  model  is  generally  well  known,  as  is  the  state  of  unbalance, 
so  the  main  source  of  error  in  the  estimates  is  due  to  measurement  noise 
and  bearing  uncertainty.  If  the  least  squares  problem  is  formulated  as 


Ax  =  b  (11) 

then 

(Aq  +  Ajv)ic  =  bo  +  b/i/  (12) 

where  the  subscript  N  refers  to  noise  and  0  to  data  which  is  noise-free.  The 
least-squares  estimate  is  given  by 

X  =  (Aq  Ao  +  A^Aiv  +  +  A^Ao)  ^(Aq  +  Ayv)^(bo  +  b;v)  (13) 

Even  if  the  noise  on  the  outputs  is  uncorrelated  with  the  noise  on  the  inputs 
the  expected  value  of  x  does  not  equal  that  of  its  estimate: 

E[x]  -  E[x]  0  (14) 

In  other  words  the  estimate  x  is  biased  [6].  In  order  to  reduce  the  bias  of 
the  estimates,  the  instrumental  variables  method  can  be  used.  Essentially, 
it  requires  the  use  of  a  matrix  that  is  uncorrelated  with  the  noise  on  the 
outputs,  but  which  is  strongly  correlated  with  the  noise-free  measurements 
themselves.  If  W  is  the  instrumental  variable  matrix,  then 

W'^Ax  =  W'^b  (15) 

Expanding 

i  =  (W^Ao  +  +  b^)  (16) 

This  means  that  E[x]  =  x,  in  other  words  unbiased  estimates  result. 
Fritzen[7]  suggested  an  iterative  method  for  solving  for  the  parameters. 
Initially,  equation  (10)  is  solved  in  a  least-squares  sense,  and  the  values 
of  the  estimated  parameters  are  used  to  calculate  outputs  for  the  model. 
These  outputs  are  then  used  to  create  W  in  the  same  way  as  the  original 
outputs  were  used  to  create  A,  new  estimates  are  obtained,  and  if  neces¬ 
sary  the  process  is  repeated.  Experience  seems  to  suggest  good  convergence 
properties[7]. 


3  SIMULATION 

The  method  under  discussion  in  this  paper  was  tested  on  a  model  of  a 
small  test  rig  located  at  Aston  University,  Birmingham.  This  consists  of 
a  steel  shaft  approximately  1.1m  long,  with  nominal  diameter  38mm.  The 
shaft  is  supported  at  either  end  by  a  journal  bearing  of  diameter  100mm, 


1063 


1  2  3  4  5  6  7  8  9  10  11  12 


Figure  2:  Rotor-bearing  system 

L/D  ratio  of  0.3  and  clearance  of  25/.tm.  There  are  two  shrink-fitted  bal¬ 
ancing  discs  for  balancing  runs.  Each  bearing  is  supported  on  a  flexible 
pedestal  to  simulate  the  flexible  foundations  encountered  in  power  station 
turbines.  At  present  these  pedestals  are  bolted  onto  a  massive  lathe  bed. 
The  rotor  is  powered  by  a  DC  motor  attached  via  a  belt  to  a  driving  pulley, 
which  is  in  turn  attached  via  a  flexible  coupling  to  the  main  rotor  shaft. 

A  schematic  of  the  rig  is  shown  in  figure  2.  Dimensions  of  each  station 
and  material  properties  are  given  in  table  1.  A  finite  element  model  of  the 
rotor  with  23  elements  was  created  and  short  bearing  theory  was  used  to 
obtain  values  for  the  bearing  stiffness  and  damping[8]. 

The  pedestals  themselves  consist  of  two  rectangular  steel  plates,  600mm 
X  150mm  which  have  two  channels  cut  into  them,  and  which  are  supported 
on  knife-edges  (figure  3).  The  vertical  stiffness  arises  from  the  hinge  effect 
of  the  channels,  whilst  the  horizontal  stiffness  is  as  a  result  of  the  shaft 
centre  tilting  under  an  applied  load.  Treating  the  supports  as  beams,  the 
theoretical  stiffnesses  are: 

Ky  =  0.^5MN/m  !<:,  =  1.5MiV/m 

where  x  and  y  refer  to  the  horizontal  and  vertical  directions  respectively. 
The  masses  and  damping  factors  were  taken  as: 

il/4  =  My  =  50/cp  Ca:  =  Cy  =  150iV  •  s/m 

The  estimation  theory  was  tested  using  this  model.  The  finite  element 
model  was  used  to  generate  responses  at  the  bearings  for  frequencies  from 


1064 


Table  1:  Table  of  rotor  rig  properties 


Shaft  Properties 

Station 

Length  (mm) 

Diameter  (mm) 

E  (GPa) 

P  (kg/m^) 

1 

6.35 

38.1 

200 

7850 

2 

25.4 

77.57 

200 

7850 

3 

50.8 

38.1 

200 

7850 

4 

203.2 

100 

200 

7850 

5 

177.8 

38.1 

200 

7850 

6 

50.8 

116.8 

200 

7850 

7 

76.2 

38.1 

200 

7850 

8 

76.2 

109.7 

200 

7850 

9 

76.2 

38.1 

200 

7850 

10 

50.8 

102.9 

200 

7850 

11 

177.8 

38.1 

200 

7850 

12 

203.2 

100 

200 

7850 

Balancing  discs 

Station 

Length  (mm) 

Diameter  (mm)  Unbalance  (kg  •  m) 

6 

25.4 

203.2 

0.001 

25.4 

203.2 

0.001 

0  to  30  Hz  with  a  spacing  of  0.1  Hz.  The  responses  were  corrupted  by 
normally  distributed  random  noise  with  zero  mean  and  standard  deviation 
of  0.1%  of  the  maximum  response  amplitude  (applied  to  both  real  and 
imaginary  parts  of  the  response).  At  each  frequency  the  bearing  static 
forces  were  disturbed  by  noise  drawn  from  a  uniform  distribution  spanning 
an  interval  of  20%  of  the  force  magnitude,  to  introduce  uncertainty  into  the 
bearing  parameters.  The  unbalance  was  assumed  to  be  exactly  known.  A 
series  of  30  runs  was  performed,  foundation  parameter  estimates  calculated 
and  the  mean  and  standard  deviations  of  these  estimates  obtained. 

The  magnitudes  of  the  responses  at  both  bearings  are  given  in  figure  4, 
which  show  that  there  are  four  critical  speeds  in  the  frequency  range  under 
consideration.  A  sample  of  true  and  estimated  forces  in  the  bearings  are 
shown  in  figures  5  and  6. 

The  means  {fi)  and  standard  deviations  (a)  of  the  least-square  (LS) 
and  instrumental  variable  (IV)  estimates  for  the  foundation  parameters 
are  shown  in  table  2. 


Displacement  (m) 


Figure  3:  Flexible  bearing  supports 


Figure  4:  Magnitudes  of  responses  at  bearings 


1066 


Table  2;  Parameter  estimates  for  foundations  showing  uncertainty 


1069 


4  DISCUSSION 


The  results  in  table  2  show  a  clear  improvement  in  parameter  estimates 
when  the  instrumental  variable  method  is  used.  There  is  a  clear  bias  in 
the  least-squares  estimates  which  is  significantly  less  when  the  instrumen¬ 
tal  variable  method  is  employed.  Also,  despite  the  fact  that  the  bearing 
parameters  are  assumed  to  be  seriously  in  error,  the  estimates  appear  to  be 
insensitive  to  them.  This  will  be  true  provided  that  the  bearings  are  much 
stiffer  than  the  foundation  (a  reasonable  assumption  in  practice).  It  does 
appear  however  that  in  some  cases  the  standard  deviation  of  the  instru¬ 
mental  variable  estimate  is  larger  than  that  of  the  least-squares  estimate, 
a  fact  which  warrants  further  investigation. 

As  far  as  the  rotor  model  is  concerned,  impact  tests,  which  are  per¬ 
formed  on  rotors  prior  to  them  entering  service,  normally  give  experimental 
frequencies  which  are  within  a  few  percent  of  the  theoretical  ones.  Thus 
the  assumption  that  the  rotor  model  is  accurately  known  would  appear  to 
be  reasonable. 

The  state  of  unbalance  may  in  theory  be  established  from  a  balancing 
run.  If  two  successive  run-downs  are  performed,  one  due  to  the  unknown 
system  unbalance  and  one  with  known  balance  weights  attached,  then  pro¬ 
vided  the  system  is  linear  the  response  measurements  may  be  vectorially 
subtracted  to  give  the  response  due  to  the  known  balance  weights  alone. 
In  order  to  ascertain  the  effect  of  unbalance  uncertainty  on  the  parameter 
estimates,  one  run  was  performed  assuming  no  error  in  the  unbalance.  It 
should  be  noted  that  this  assumes  that  the  system  is  linear. 

5  CONCLUSIONS 

A  method  of  estimating  turbogenerator  foundation  parameters  from 
potentially  noisy  measurement  data  is  demonstrated.  It  is  shown  that 
making  use  of  the  instrumental  variable  method  reduces  the  bias  in  the 
estimates  and  improves  them  quite  significantly. 

6  ACKNOWLEDGEMENT 

The  authors  wish  to  acknowledge  the  support  and  funding  of  Nuclear 
Electric  Ltd  and  Magnox  Electric  Pic.  Dr  Friswell  wishes  to  acknowledge 
the  support  of  the  Engineering  and  Physical  Sciences  Research  Council 
through  the  award  of  an  advanced  fellowship. 


1070 


REFERENCES 


[1]  A.  W.  Lees  and  I.  C.  Simpson.  The  dynamics  of  turbo-alternator  foun¬ 
dations:  Paper  C6/83.  In  Conference  on  steam  and  gas  turbine  founda¬ 
tions  and  shaft  alignment,  Bury  St  Edmunds,  1983,  IMechE,  pp37-44. 

[2]  A.  W.  Lees.  The  least  squares  method  applied  to  identify  ro¬ 
tor/foundation  parameters:  Paper  C306/88.  In  Proceedings  of  the  Inter¬ 
national  Conference  on  Vibrations  in  Rotating  Machinery,  Edinburgh, 
1988,  IMechE,  pp209-216. 

[3]  M.  1.  Friswell  and  A.  W.  Lees.  Estimation  of  forces  exerted  on  machine 
foundations.  In  M.  1.  Friswell  and  J.  E.  Mottershead,  editors,  Interna¬ 
tional  Conference  on  Identification  in  Engineering  Systems,  Swansea, 
1996,  pp793-803. 

[4]  N.  S.  Feng  and  E.  J.  Hahn.  Including  foundation  effects  on  the  vibra¬ 
tion  behaviour  of  rotating  machinery.  Mechanical  Systems  and  Signal 
Processing,  1995,  9,  pp243-256. 

[5]  G.  A.  Zanetta.  Identification  methods  in  the  dynamics  of  turbogener¬ 
ator  rotors:  Paper  C432/092.  In  IMechE  Conference  on  Vibrations  in 
rotating  machinery,  Bath,  1992.  IMechE,  ppl73-181. 

[6]  J.  Schoukens  and  R.  Pintelon.  Identification  of  linear  systems.  Perga- 
mon  Press,  1991. 

[7]  C.  P.  Fritzen.  Identification  of  mass,  damping  and  stiffness  matrices  of 
mechanical  systems.  Journal  of  Vibration,  Acoustics,  Stress  and  Relia¬ 
bility  if  Design,  1986  108,  pp9-17. 

[8]  D.  M.  Smith.  Journal  bearings  in  Turbomachinery.  Chapman  and  Hail, 
1969. 


1071 


1072 


SHELL  MODE  NOISE  IN  RECIPROCATING  REFRIGERATION 
COMPRESSORS 


Ciineyt  Oztiirk  and  Aydin  Bahadir 
Tiirk  Elektrik  Endiistrisi  A.$ 

R&D  Department 

Davutpa§a,  Litres  Yolu,  Topkapi  -34020,  Istanbul,  Turkey 


ABSTRACT 

This  study  describes  the  successful  endeavor  to  understand  the  causes  of  noise 
that  appear  on  the  shell  modes  of  the  reciprocating  refrigeration  compressors. 
The  compressor  shell  is  generally  considered  as  the  acoustic  enclosure  that 
reflects  the  acoustic  energy  back  into  the  compressor  cavity  but  also  as  the 
transmitter  and  radiator  of  the  transmitted  acoustic  energy  that  could  be 
radiated  into  the  air  or  transmitted  to  the  structure.  Vibrations  of  the 
compressor  shell  can  easily  be  characterized  in  terms  of  the  modal  parameters 
that  consist  of  the  natural  frequencies,  mode  shapes  and  damping  coefficients. 
The  noise  source  harmonics  and  the  shell  resonances  couple  to  produce  the 
shell  noise  and  vibration.  The  harmonic  spacing  is  equal  to  the  basic  pumping 
frequency.  Results  of  the  studies  indicate  that  important  natural  frequencies  of 
the  compressor  shell  usually  stay  between  2000-6000  Hz  interval.  The 
important  natural  frequencies  are  first  natural  frequencies  in  the  lower  range 
with  the  longer  wavelengths  that  radiate  well. 


INTRODUCTION 

Compressor  noise  sources  are  those  processes  where  certain  portions  are 
separated  from  the  desired  energy  flow  and  transmitted  through  the  internal 
components  of  compressor  to  the  hermetic  shell  where  it  is  radiated  from  the 
shell  as  airborne  noise  on  vibration  of  supporting  structure  will  eventually 
radiate  noise  from  some  portion  of  the  structure.  Noise  sources  of  the 
reciprocating  refrigeration  compressors  can  be  classified  as  motor  noise, 
compression  process  noise  and  valve  port  flow  noise. 


1073 


In  reciprocating  compressors  there  is  very  high  density  of  noise  harmonics  even 
though  they  decay  in  amplitude  at  high  frequencies.  Generally,  these  noise 
source  harmonics  and  the  shell  resonances  couple  to  produce  shell  noise  and 
vibration 


NOISE  GENERATION  MECHANISM  OF  THE  RECIPROCATING 
REFRIGERATION  COMPRESSORS 

Significance  of  the  problem 

The  results  of  the  sound  radiation  characteristics  shown  at  figure  1  indicated 
that  certain  high  amplitude  frequency  components  had  very  distinctive  sound 
radiation  patterns.  It  was  suspected  these  frequencies  correspond  to  excitation 
of  either  structural  resonances  of  the  compressor  shell  or  acoustic  resonances  of 
the  interior  cavity  space.  Resonances  those  amplify  the  noise  and  vibration 
caused  by  pumping  harmonics  of  a  compressor  and  thus  can  be  the  cause  of 
significant  noise  problems. 


Sound  power  -  A-Wcighted 


— I - , - 1 - , - ; - P — — r — f — ! — — 'I'  "I  '"'I  — I - ”r '  i  i 

50.0  63.0  80,0  100,0  125,0  160,0  200,0  250,0  315,0  400,0  500,0  630,0  800,0  l,0k  t,3k  1,6k  2,0k  2,5k  4,0k  S,0k  63k  S,0k  10,0k 

Frequency  [Hz] 


Figure  1,  Noise  Radiation  Characteristics  of  Reciprocating  Refrigeration 
Compressor. 

Noise  sources 

Noise  in  a  compressor  is  generated  during  cyclic  compression,  discharge, 
expansion  and  suction  process.  The  character  of  noise  sources  is  harmonic  due 
to  periodic  nature  of  the  compression  process.  These  harmonics  are  present  in 
the  compression  chamber,  pressure  time  history  and  loading  of  the  compressor 
through  drive  system.  The  motor  can  not  provide  immediate  response  to  load 
harmonics  and  load  balance  is  obtained  at  the  expense  of  acceleration  and 


1074 


deceleration  of  the  motor  drive  system.  Harmonic  vibrations  of  the  motor  drive 
system  can  then  excite  the  resonant  response  in  the  compressor  components 
that  can  transmit  the  acoustic  energy  in  very  efficient  way.  The  rest  of  the  noise 
sources  are,  turbulent  nature  of  flow  depending  to  passage  through  valve  ports, 
valve  impacts  on  their  seats  and  possible  amplification  when  matched  with 
mechanical  resonances. 

MECHANICAL  FEATURES  OF  SHELL 
Mechanical  features  of  the  compressor 

The  hermetically  sealed  motor  compressor  comprises  in  general  a  motor 
compressor  unit  including  a  motor  assembly  mounted  with  a  frame  and  a  sealed 
housing  within  which  the  compressor  is  supported  by  means  of  plurality  of  coil 
springs  each  having  one  end  spring  with  the  frame  and  the  other  end  connected 
with  the  interior  of  the  housing. 

The  refrigerant  gas  as  it  is  compressed  in  the  cylinder  is  discharged  through  the 
discharge  chamber  in  the  cylinder  head  into  the  discharge  muffler.  The 
discharge  muffler  is  generally  mounted  on  the  cylinder  head  attached  in 
covering  relation  to  an  end  face  of  the  cylinder.  Where  the  sealed  casing  is 
spherical  in  shape  for  better  noise  suppression,  an  upper  end  of  the  cylinder 
head  tends  to  interfere  with  an  inner  wall  surface  of  the  casing,  a  disadvantage 
that  can  only  be  eliminated  by  increasing  the  size  of  scaled  casing  for  providing 
a  desired  hermetically  sealed  motor. 

Compressor  Shell 

The  shell  is  easily  be  characterized  with  the  modal  parameters.  The  ideal  shell 
should  be  designed  in  a  way  that  keep  all  the  excitation  frequencies  at  the  mass 
controlled  region  of  all  its  modes.  But,  depending  on  the  very  tight  constraints 
that  come  with  the  gas  dynamics  and  motor  locations,  it  is  not  allowed  to  be 
flexible  during  the  design  of  shell.  As  a  consequence  of  the  existing  design 
limitations  natural  frequencies  of  the  shell  usually  fall  between  1000-5000  Hz. 

SHELL  RADIATION 

Figure  1  illustrates  how  the  sound  pressure  level  of  a  pumping  harmonic  can 
increase  as  it  nears  a  resonant  frequency.  The  sound  pressure  level  of  the 
pumping  harmonic  increases  around  the  shell  resonances.  The  resonance 
generally  radiates  primarily  from  the  large  flat  sides  of  the  compressor.  There 
are  three  major  acoustic  cavity  resonances  400,  500  and  630  Hz  and  four  major 
structural  resonances  of  the  compressor  shell:  around  1.6  K,  2  K,  2.5  and  3.2 
K.  Hz  at  which  noise  radiates  well  in  certain  directions. 


1075 


SHELL  RESONANCES 


Figures  2,  3  and  4  illustrate  the  frequency  responses  of  the  compressor  shell 
when  measured  with  the  impact  hammer  method.  Figure  2  is  the  response  to 
the  excitation  in  x  direction,  figure  3  is  for  the  excitation  direction  and  figure  4 
is  for  the  excitation  in  z  direction. 


Figure  2,  Frequency  response  of  the  compressor  shell  when  excited  in  X 
direction. 


Figure  3,  Frequency  response  of  the  compressor  shell  when  excited  in  Y 
direction. 


1076 


Figure  4,  Frequency  response  of  the  compressor  shell  when  excited  in  Z 
direction. 


To  verify  the  hypothesis  that  resonances  were  contributing  to  some  of  the  noise 
problems  of  the  reciprocating  piston  compressor,  a  modal  analysis  of  the  shell 
and  interior  cavity  was  performed. 

For  the  shell  modal  analysis,  the  accelerometer  to  measure  the  response 
remained  stationary,  while  the  impact  location  was  moved.  The  test  was 
performed  in  this  manner  for  convenience  since  it  was  easier  to  fix  the 
accelerometer  in  one  location  and  strike  the  compressor  with  force  hammer  at 
each  grid  point  to  obtain  transfer  function  for  each  measurement  location. 
Identical  results  are  obtained  if  impact  occurred  at  a  single  point  and  the 
response  was  measured  at  each  measurement  location.  Preliminary  test  were 
performed  initially  to  identify  an  appropriate  measurement  location  at  which  all 
important  natural  frequencies  of  shell  are  detectable.  Several  force  input  and 
response  locations  were  evaluated  to  determine  the  best  location  to  mount 
accelerometer  to  measure  the  shell  response. 

The  shell  resonances  are  also  calculated  by  using  the  Structural  FEM  analysis. 
Table- 1  lists  the  natural  frequencies  predicted  in  these  studies.  During  the  finite 
element  analysis,  the  models  of  the  compressor  were  built,  based  on  the  CAD 
models.  The  shell  consists  of  7500  elements.  The  mesh  densities  are  quite 
adequate  for  the  structural  analysis  in  the  frequency  range  of  interest.  In  order 
to  investigate  the  possible  influence  of  the  crank  mechanism  on  the  natural 
frequencies  of  the  shell,  a  simple  model  of  the  crank  mechanism  was  introduced 
to  the  FE  model.  During  the  calculations,  the  crank  mechanism  was  simplified 
as  a  rigid  block  with  certain  mass  and  rotary  inertia  and  modeled  with  solid 
elements.  The  shell  and  the  crank  mechanism  have  been  suspended  with  the 


1077 


springs  from  3  positions  and  in  all  6  transitional  and  rotational  directions. 
Depending  on  the  negligible  spring  effect  on  the  longitudinal  direction, 
estimated  values  have  been  used  in  5  directions.  The  FE  model  has  been 
assumed  to  be  free-free. 


Mode 

# 

Frequency 

Hz 

Mode 

# 

Frequency 

Hz 

1 

1997 

21 

4716 

2 

22 

4925 

3 

2293 

23 

4994 

4 

24 

5 

25 

5159 

6 

26 

5454 

7 

2889 

27 

5476 

8 

3258 

28 

9 

29 

EESI^HI 

10 

30 

5783 

11 

3376 

31 

5936 

12 

3551 

32 

5999 

13 

3577 

33 

6035 

14 

34 

6055 

15 

3788 

35 

6183 

16 

3958 

36 

6237 

17 

4383 

37 

6314 

18 

4481 

38 

19 

4644 

39 

20 

4702 

40 

6701  1 

Table- 1,  Calculated  Natural  Frequencies  of  the  Shell 

Figures  5,  6,  7  and  8  illustrate  how  the  mode  shapes  of  the  shell  vary  at  the 
mode  frequencies  of  2754,  3332,  3551  and  3788  Hz  ,  These  figures  indicate 
that  the  shell  vibrates  predominantly  along  the  large  flat  sides  of  the  compressor 
at  points  where  the  suspension  springs  are  attached  to  the  shell  wall  at  these 
natural  frequencies.  When  referred  to  figure  1  of  the  noise  radiation  this 
frequency  range  is  also  the  range  where  the  noise  radiates  efficiently  from  the 
large  flat  sides  of  the  compressor.  Thus,  there  is  good  correlation  between  the 
acoustic  data  and  structural  data  for  these  frequencies.  The  slight  discrepancies 
in  the  structural  natural  frequencies  and  the  acoustic  data.  Acoustic  data  have 
been  picked  up  at  the  shell  temperature  of  the  reciprocating  piston  compressor 
that  could  reach  up  to  1 10  C. 


The  modal  analysis  results  also  indicate  that  the  compressor  suspension  springs 
are  attached  to  a  point  on  the  shell  where  the  shell  is  comparatively  compliant. 
Thus,  the  vibrational  energy  transmitted  through  the  springs  to  the  compressor 
shell  can  and  did  effectively  excite  the  shell  vibrations.  Also,  significant  shell 
vibrations  occur  along  the  large  flat  sides  of  the  compressor  shell  indicating  the 
curvature  of  the  shell  needs  to  be  increased  to  add  stifihiess  to  the  shell. 

Based  on  the  results  of  the  shell  modal  analysis,  it  is  recommended  the 
suspension  springs  moved  away  from  the  compliant  side  walls  of  the  shell.  A 
four  spring  arrangement  at  the  bottom  of  the  shell  near  corners  where  the 
curvature  is  sharp  would  reduce  the  amount  of  vibration  energy  transferred  to 
the  shell  because  of  the  reduced  input  mobility  of  the  shell  at  these  locations. 

It  is  also  believed  increasing  the  stiffness  of  the  shell  by  increasing  the  curvature 
will  provide  noise  reduction  benefits.  The  greater  shell  stiffness  lowers  the 
amplitude  of  the  shell  vibrations.  Figure  9,  illustrate  the  third  octave  change  in 
compressor  noise  with  the  same  compressor  in  the  new  shell.  An  over  all  noise 
level  of  5  dBA  has  been  obtained. 


Figure  9  Compressor  noise  level  improvement  after  the  shell  modification. 

The  increased  shell  stiffness  also  raises  the  natural  frequencies  of  the  shell 
where  there  is  less  energy  for  transfer  function  response.  However,  there  is  a 
possible  disadvantage  to  increasing  stiffness  of  the  shell.  The  higher  natural 
frequency  lowers  critical  frequency  of  the  shell  thus  reducing  transmission  loss 
of  the  shell. 


1081 


Damping  treatments  can  also  have  obvious  benefits  in  vibration  and  noise 
reduction.  Visco  elastic  and  Acoustic  dampings  are  considered  to  avoid  the 
shell  excitations.  The  application  of  dampers  can  also  provide  up  to  5  dBA 
reductions  when  appropriately  located  on  the  shell. 

CONCLUSION 

The  results  of  studies  indicate  that  structural  resonances  of  the  shell  are  indeed 
amplifying  the  noise  due  to  the  pumping  harmonics  of  the  reciprocating 
refrigeration  compressor  to  cause  significant  noise  radiation  outside  of 
compressor. 

In  order  to  tackle  with  this  noise  problem,  within  the  scope  of  these  studies 
two  different  effective  shell  noise  control  are  considered  based  on  the  results  of 
numerical  and  experimental  structural  analysis  and  acoustic  features  of  the 
reciprocating  compressor.  First,  shell  noise  control  method  is  the  redesign  of 
the  shell  with  increased  stiffness  by  replacing  all  the  abrupt  changes  in  the 
curvature  with  the  smooth  continuous  changes.  It  is  apparent  that  change  in  the 
shell  configuration  can  shift  the  first  shell  resonance  from  1750  Hz  to  nearly 
3200  Hz.  The  results  of  the  redesign  effects  can  reach  up  to  3-5  dBA  reduction 
on  third  octave  noise  levels.  Second  treatment  that  could  be  applied  against  the 
excitation  of  shell  resonances  are  considered  as  the  acoustic  and  viscoelastic 
dampers.  These  dampers  can  be  chosen  to  operate  efficiently  at  the  shell 
resonant  frequencies.  These  two  applications  can  also  provide  up  to  2-5  dBA 
reduction  on  the  third  octave  band  of  the  noise  emission  but  the  long  term 
endurance  and  temperature  dependence  of  these  materials  can  always  be  a 
question  mark  when  considered  from  the  manufacturer  point  of  view. 


REFERENCES 

1.  JFROBATTAand  ID.  JONES  1991,  Purdue  University,  School  of 
Mechanical  Engineering,  The  Ray  Herrick  Laboratories,  Report  no :  1912-1 
HL  91-9P,  73-84,  Investigation  of  Noise  Generation  Mechanisms  and 
Transmission  Paths  of  Fractional  Horsepower  Reciprocating  Piston  and 
Rolling  Piston  Compressors 

2.  HAMILTON  I F  1988,  Purdue  University ,  School  of  Mechanical 
Engineering,  The  Ray  Herrick  Laboratories,  207-213  Measurement  and 
Control  of  Compressor  Noise 

3.  C  OZTURK,  A  AQIKGOZ  and  J  L  MIGEOT  1996,  International 
Compressor  Engineering  Conference  at  Purdue,  Conference  Proeceeding , 
Volume  II,  697-703,  Radiation  Analysis  of  the  Reciprocating  Refrigeration 
Compressor  Casing 


1082 


A  COMPARATIVE  STUDY  OF  MOVING  FORCE  IDENTIFICATION 


T.H.T.  Chan,  S.S.  Law,  T.H.  Yung 

Department  of  Civil  &  Structural  Engineering, 

The  Hong  Kong  Polytechnic  University,  Hung  Horn,  Kowloon,  Hong  Kong 


ABSTRACT 

Traditional  ways  to  acquire  truck  axle  and  gross  weight  information  are 
expensive  and  subject  to  bias,  and  this  has  led  to  the  development  of  Weigh- 
in-Motion  (WIM)  techniques.  Most  of  the  existing  WIM  systems  have  been 
developed  to  measure  only  the  static  axle  loads.  However  dynamic  axle  loads 
are  also  important.  Some  systems  use  instrumented  vehicles  to  measure 
dynamic  axle  loads,  but  are  subject  to  bias.  These  all  prompt  the  need  to 
develop  a  system  to  measure  the  dynamic  axle  loads  using  an  unbiased 
random  sample  of  vehicles.  This  paper  aims  to  introduce  four  methods  in 
determining  such  dynamic  forces  from  bridge  responses.  The  four  methods  are 
compared  with  one  another  based  on  maximum  number  of  forces  to  be 
identified,  minimum  number  of  sensors,  sensitivity  towards  noise  and  the 
computation  time.  It  is  concluded  that  acceptable  estimates  could  be  obtained 
by  all  the  four  methods.  Further  work  includes  merging  the  four  methods  into 
a  Moving  Force  Identification  System  (MFIS). 


INTRODUCTION 

The  truck  axle  and  gross  weight  information  have  application  in  areas 
such  as  the  structural  and  maintenance  requirements  of  bridges  and  pavements. 
However,  the  traditional  ways  to  acquire  that  are  expensive  and  subject  to  bias, 
and  this  has  led  to  the  development  of  Weigh-in-Motion  (WIM)  techniques. 
Some  systems  are  road-surface  systems  which  make  use  of  piezo-electric 
(pressure  electricity)  or  capacitive  properties  to  develop  a  plastic  mat  or 
capacitive  sensors  to  measure  axle  weight  [1].  Another  kind  of  WIM  system  is 
the  under-structure  systems  in  which  sensors  are  installed  under  a  bridge  or  a 
culvert  and  the  axle  loads  are  computed  from  the  measured  responses  e.g. 
AXWAY  [2]  and  CULWAY  [3].  All  the  above  mentioned  systems  can  only 
give  the  equivalent  static  axle  loads.  However  dynamic  axle  loads  are  also 
important  as  they  may  increase  road  surface  damage  by  a  factor  of  2  to  4  over 
that  caused  by  static  loads  [4].  Some  systems  use  instrumented  vehicles  to 
measure  dynamic  axle  loads  [5],  but  are  subject  to  bias.  These  all  prompt  the 
need  to  develop  a  system  to  measure  the  dynamic  axle  loads  using  unbiased 
random  samples  of  vehicles.  Four  methods  are  developed  to  determine  such 


1083 


dynamic  forces  from  bridge  responses  which  include  bending  moments  or 
accelerations. 


EQUATION  OF  MOTIONS  FOR  MOVING  LOADS 

The  moving  force  identification  methods  described  in  this  paper  are  the 
inverse  problems  of  an  predictive  analysis  which  is  defined  by  0‘ Connor  and 
Chan  [6]  as  an  analysis  to  simulate  the  structural  response  caused  by  a  set  of 
time-varying  forces  running  across  a  bridge.  Two  models  can  be  used  for  this 
kind  of  analysis. 


A.  Beam-Elements  Model 


0‘ Connor  and  Chan  [6]  model  the  bridge  as  an  assembly  of  lumped 
masses  interconnected  by  massless  elastic  beam  elements  as  shown  in  Figure  1, 
and  the  nodal  responses  for  displacement  or  bending  moments  at  any  instant 
are  given  by  Equations  (1)  and  (2)  respectively. 


Moving  Loads 


...  ip.  Ip,  ^ 

- D — D — 2^ 


Lumped  Masses  1  2  ...  N-1  N 


Figure  1  -  Beam-Elements  Model 


{Y}  =  [Y,]{F}-lY,][Am]{n-m[C]{Y}  (1) 

{5M}  =  [5MJ{P}-[5M;][Am]{y}-[5MJ[C]{f}  (2) 

where  [P]  is  a  vector  of  wheel  loads,  [Am]  is  a  diagonal  matrix  containing 

values  of  lumped  mass,  [C]  is  the  damping  matrix,  5M,L,7,7are  the  nodal 
bending  moments,  displacements,  velocities  and  accelerations  respectively, 
[R,^]  {'R  ’  can  be  Y  or  BAP)  is  an  m  x  n  matrix  with  the  ith  column  representing 
the  nodal  responses  caused  by  a  unit  load  acting  at  the  position  of  the  ith  wheel 
load  and  [i?J  {'R'  can  be  Y  or  BM)  is  an  n  x  n  matrix  with  the  ith  column 
representing  the  nodal  responses  caused  by  a  unit  load  acting  at  the  position  of 
ith  internal  node. 


1084 


B.  Continuous  Beam  Model 


Assuming  the  beam  is  of  constant  cross-section  with  constant  mass  per 
unit  length,  having  linear,  viscous  proportional  damping  and  with  small 
deflections,  neglecting  the  effects  of  shear  deformation  and  rotary  inertia 
(Bernoulli-Euler’s  beam),  and  the  force  is  moving  from  left  to  right  at  a 
constant  speed  c,  as  shown  in  Figure  2,  then  the  equation  of  motion  can  be 
written  as 


P 


d-v{xj) 

a- 


a 


+  EI 


^v(x,0 

a- 


S(x-ct)f{t) 


(3) 


where  v(x,0  is  the  beam  deflection  at  point  x  and  time  t;  p  is  mass  per  unit 
length;  C  is  viscous  damping  parameter;  £  is  Young's  modulus  of  material;  I 
is  the  second  moment  of  inertia  of  the  beam  cross-section;  f(t)  is  the  time- 
varying  force  moving  at  a  constant  speed  of  c,  and  Sft)  is  Dirac  delta  function. 


\/(0 


— O 


Figure  2  -  Simply  supported  beam  subjected  to  a  moving  force  f(t) 

Based  on  modal  superposition,  the  dynamic  deflection  y(x,t)  can  be 
described  as  follows; 


v{x,t)  =  Y,<i'„{x)q„{t) 

}}=\ 


(4) 


where  n  is  the  mode  number;  0n(x^)  is  the  mode  shape  function  of  the  n-th 
mode  and  qyi(t)  is  the  n-th  modal  amplitudes. 

Based  on  the  above  mentioned  predictive  analyses,  four  Moving  Force 
Identification  Methods  (MFIM)  are  developed. 


1085 


FIRST  MOVING  FORCE  IDENTIFICATION  METHOD 
INTERPRETIVE  METHOD  I  -  BEAM-ELEMENTS  MODEL  (IMI) 


It  is  an  inverse  problem  of  the  predictive  analysis  using  beam-elements 
model  From  Equation  (1),  it  can  be  seen  that  if  Y  is  known  at  all  times  for  all 
interior  nodes,  then  7  and  Y  can  be  obtained  by  numerical  differentiation. 
Equation  (1)  becomes  an  overdetermined  set  of  linear  simultaneous  equations 
in  which  the  P  may  be  solved  for  them.  However  a  particular  difficulty  arises 
if  measured  BM  are  used  as  input  data.  Remembering  that  the  moving  loads  P 
are  not  normally  at  the  nodes,  the  relation  between  nodal  displacements  and 
nodal  bending  moments  is 

{r}^[Y,]{BM}  +  [Yc]{B}  (5) 

where  allows  for  the  deflections  due  to  the  additional  triangular 

bending  moment  diagrams  that  occur  within  elements  carrying  one  or  more 
point  loads  P.  [}^]can  be  calculated  from  the  known  locations  of  the  loads. 
[}^^]  and  {BM}aic  known,  but  {7}  cannot  be  determined  without  a 
knowledge  of  (P}.  0‘Connor  and  Chan  [6]  describe  a  solution  uses  values  of 
{P}  assumed  from  the  previous  time  steps. 


SECOND  MOVING  FORCE  IDENTIFICATION  METHOD 
INTERPRETIVE  METHOD  II  -  CONTINUOUS  BEAM  MODEL  (IMII) 

From  the  predictive  analysis  using  continuous  beam  model,  if  the  ith- 

iTTX 

mode  shape  function  of  the  simply  supported  Euler‘s  beam  is  sin-—-,  then 
the  solution  of  Equation  (3)  takes  the  form 

v  =  ^sin— F;(0  (6) 

/=i  L 

where  V.(t) ,  (i  =  1,  2,  •  •  •)  are  the  modal  displacements. 

Substitute  Equation  (6)  into  Equation  (3),  and  multiply  each  term  of 
Equation  (3)  by  the  mode  shape  function  sin(y;r.x  /  L) ,  and  then  integrate  the 
resultant  equation  with  respect  to  x  between  0  and  L  and  use  the  boundary 
conditions  and  the  properties  of  Dirac  function.  Consequently,  the  following 
equation  can  be  obtained 


1086 


op 


u  2  El  ^ 

where  =  —5 — -  C(y,  = 


C 


2/rryy, 


at  the  j-th  mode. 


If  there  are  more  than  one  moving  loads  on  the  beam,  Equation  (7)  can  be 
written  as 


.  7r(ct-Xk) 

._7r(ct~x,) 

7r{ct-x^)  ■ 

■^.1 

'2CM' 

'cojv; 

sin - ^ 

L 

sin - ^  • 

L 

sin - — 

L 

'K 

V, 

2C,co,K 

+ 

CO  IVj 

_  2 

.  27r(ct-Xt) 
sin 

.  27r(ct~x^} 
sm - =-  ■ 

.  27r{ct  -  x. ) 

••  sm — - — 

Pi 

~  Ml 

A 

.1 

_2C,.a)/„_ 

_coX_ 

.  n7r(ct-x,) 
sin — ^ ^ 

.  riKict-x^) 
sin — ^ - —  . 

.  n7c{ct-x^) 

••  sin — - — 

A. 

L 

L 

L 

(8) 


in  which  is  the  distance  between  the  k-th  load  and  the  first  load  and  x,  =  0 . 


If  Pi  Pk  are  constants,  the  closed  form  solution  of  Equation  (3)  is 

/  N  ^  1  •  ■  j7E{ct-Xi)  a  .  .  .  ,  ^  {Q\ 

48E/f:i'  L  \  L  j  ^  J 

nc 


in  which  a  = 


Lcd 


If  we  know  the  displacements  of  the  beam  at  x, ,  x, ,  •  •  • ,  ,  the  moving 

loads  on  the  beam  are  given  by 

{v}  =  [S, „.]{/>}  (10) 

in  which  {v}=[v|  v,  ■■■  v,]'  {•^’}  =  [-^1  ^2  ■" 


^n- 


,  where 


L"  A  1  .  j7rx,J  .  J7r{ct-x.)  a  .  ,  .  ,  , 

=  7577X^2772 - 27^“ “7~  - 7 - /c) 

48£:/^;-0- -a  )  L  \  L  j 


1087 


If  l>k,  that  means  the  number  of  nodal  displacements  is  larger  than  or 
equal  to  the  number  of  axle  loads,  then  according  to  the  least  squares  method, 
the  equivalent  static  axle  load  can  be  given  by 

{/>}=([s,,]''[s„])''[s,q^{v}  (11) 

If  the  loads  are  not  constant  with  time,  then  central  difference  is  used  to 
proceed  from  modal  displacements  to  modal  velocities  and  accelerations. 
Equation  (8)  becomes  a  set  of  linear  equations  in  which  P^.  for  any  instant  can 
be  solved  by  least  squares  method.  Similar  sets  of  equations  could  be  obtained 
for  using  bending  moments  to  identify  the  moving  loads. 


THIRD  MOVING  FORCE  IDENTIFICATION  METHOD 
TIME  DOMAIN  METHOD  (TDM) 


This  method  is  based  on  the  system  identification  theory  [7].  Substituting 
Equation  (4)  into  Equation  (3),  and  multiplying  each  term  by  0j(x),  integrating 
with  respect  to  x  between  0  and  L,  and  applying  the  orthogonality  conditions, 
then 


dt- 


dt 


M., 


(12) 


where  con  is  the  modal  frequency  of  the  n-th  mode;  is  the  damping  ratio  of 
the  n-th  mode;  Mn  is  the  modal  mass  of  the  n-th  modt,  pn(t)  is  the  modal  force 
and  the  mode  shape  function  can  be  assumed  as  0„(x)  =  sin(^;w  /  L) . 

Equation  (12)  can  be  solved  in  the  time  domain  by  the  convolution 
integral,  and  yields 

=  (13) 

^11  0 

where  ~  sin(^u„0,  t  >0  (14) 

and  co\,  (15) 

Substituting  Equation  (13)  into  Equation  (4),  the  dynamic  deflection  of 
the  beam  at  point  x  and  time  t  can  be  found  as 

v(x,0  =  y — ^sin^^  sin6>|,(^  -  '^)sin^^^^/(T)dr  (16) 

fxpLo),,  L  i  L 


1088 


A.  Force  Identification  from  Bending  Moments 


The  bending  moment  of  the  beam  at  point  x  and  time  t  is 


m{xj)  =  -El - 


3c- 


(17) 


Substituting  Equation  (16)  into  Equation  (17),  and  assuming  the  force  f(t) 
is  a  step  function  in  a  small  time  interval  and  f(t)  =0  at  the  entry  and  exit,  then 
let 


^  2El7r'  n~  .  yitu:  ^ 

C,„  =  — 73 - ^sm— Ar, 

pi  co„  L 

{k)  =  sin(<:y'„  A/A:), 

^  .  ,n7(cNi 

52(*)  =  sm(— ^/c) 


(18) 


(19) 


N„=- 

Equation  (17)  can  be  expressed  as 


cEt 


m(2)] 

0 

...  0  ' 

/(i) 

m(3) 

< 

■=S  c,„ 
/)  =  1 

E;,S,{2)S,i\) 

e:S,{\)S,{2) 

...  0 

/(2) 

m{N) 

_E::"S,iN-\)S,il) 

E;^--S,{N~2)S,i2) 

Kc. 

(20) 


where  A^  is  the  sample  interval  and  N+I  is  the  number  of  sample  points,  and 


f>„=£r'''“*'S,(Al-A^„+l)S,(//„-l) 


Equation  (20)  can  be  simplified  as 

B  f  =  m  (21) 

(A'-l)x(;V/^-|)  ('V/^->)xl  (,V_l)xl 

U  N  =  matrix  B  is  a  lower  triangular  matrix.  We  can  directly  find 
the  force  vector  f  by  solving  Equation  (21).  If  77  >  and/or  77/  bending 
moments  (77/  >  1)  are  measured,  least  squares  method  can  be  used  to  find 

the  force  vector  f  from 


1089 


(22) 


B,  ■ 

m, 

f 

m. 

®-v,. 

The  above  procedure  is  derived  for  single  force  identification.  Equation 
(21)  can  be  modified  for  two-forces  identification  using  the  linear 
superposition  principle  as 

“B.  0 

B,  B, 

B.  B, 

where  B 3  [N^x  (Nb-1)]  ,  Bt  [(N  -1-2NJx  (Ng-l)],  and  B^  [NjX  (Nb  -1)]  are 
sub-matrices  of  matrix  B.  The  first  row  of  sub-matrices  in  the  first  matrix 
describes  the  state  having  the  first  force  on  beam  after  its  entry.  The  second 
and  third  rows  of  sub-matrices  describe  the  states  having  two-forces  on  beam 
and  one  force  on  beam  after  the  exit  of  the  first  force. 

B.  Identification  from  Bending  Moments  and  Accelerations 


Similarly  the  acceleration  response  of  the  beam  can  be  expressed  as 

A  f  =  V 


(24) 


The  force  can  also  be  found  from  the  measured  acceleration  from 
Equation  (24).  If  the  bending  moments  and  accelerations  responses  are 
measured  at  the  same  time,  both  of  them  can  be  used  together  to  identify  the 
moving  force.  The  vector  m  in  Equation  (21)  and  v  in  Equation  (24)  should  be 
scaled  to  have  dimensionless  unit,  and  the  two  equations  are  then  combined 
together  to  give 


A/iv||J  [v/llvlj 


(25) 


where  Ihli  is  the  norm  of  the  vector. 


FOURTH  MOVING  FORCE  IDENTIFICATION  METHOD 
FREQUENCY  DOMAIN  METHOD  (FDM) 

Equation  (12)  can  also  be  solved  in  the  Frequency  Domain.  Performing 
the  Fourier  Transform  for  Equation  (12), 


1090 


1 


1 


(26) 


where 


co;,  -CO-  +24„co„co  M„ 


W 


i 

—00 


(27) 

(28) 


Let 


_ 1 _ 

co;,-co-  +2^„a„a 


(29) 


Hn(co)  is  the  frequency  response  function  of  the  n-th  mode.  Performing  the 
Fourier  Transform  of  Equation  (4),  and  substituting  Equations  (26)  and  (29) 
into  the  resultant  equation,  the  Fourier  Transform  of  the  dynamic  deflection 
v(x,t)  is  obtained  as 


A.  Force  Identification  from  Accelerations 

Based  on  Equation  (30),  the  Fourier  Transform  of  the  acceleration  of  the 
beam  at  point  .x  and  time  t  can  be  written  as 

V(x,a)  =  -®^X^(D„(x)//„(cy)P„(cy)  (31) 

Considering  the  periodic  property  of  the  Discrete  Fourier  Transform  (DFT), 
and  let 


A/^m- 


Equation  (32)  can  be  rewritten  as 


(32) 


1091 


,m  =  0,l,  (33) 


V{m)  =  X  H„(m)'i'„(m)[F,{0)  +  iF,(0)] 

/;=1 

A^/2-1  «  _ 
k=\  H=t 

Nn~\  =0  _ 

/t  =  l  /)  =  l 

+E  /2)[F„(A^  /2)  -  iivCA^  /  2)] 

«=I 

where  is  the  Fourier  Transform  of  the  n-th  mode  shape,  and  F  is  the 
Fourier  Transform  of  the  moving  force. 

Writing  Equation  (33)  into  matrix  form  and  dividing  F  and  V  into  real  and 
imaginary  parts,  it  yields 


Because  F,  (0)  =  0,  Fj  (N  /  2)  =  0,Vj  (0)  =  0,F;  ( /  2)  =  0 ,  Equation  (34)  can 
be  condensed  into  a  set  of  N  order  simultaneously  equations  as 

(35) 

Fr  and  F,  can  be  found  from  Equation  (35)  by  solving  the  Nth  order  linear 
equation.  The  time  history  of  the  moving  force  f(t)  can  then  be  obtained  by 
performing  the  inverse  Fourier  Transformation. 

If  the  DFTs  are  expressed  in  matrix  form,  the  Fourier  Transform  of  the 
force  vector  f  can  be  written  as  follows  if  the  terms  in  f  are  real  [8]. 

F  =  — Wf  (36) 

N 

whereW  =  e''^‘‘^^^  (37) 


1092 


0  0  0  0  0 

0  1  2  •••  -2  -1 

0  2  4  •••  -4  -2 

0  -2  -4  •••  4  2 

0  -1  -2  2  1 


yv  X 


The  matrix  W  is  an  unitary  matrix,  which  means 

w-‘  =(w*  y 


(38) 


where  W*  is  a  conjugate  of  W.  Substituting  Equation  (36)  into  Equation  (35), 


V  =  — Al 

N 


W  0 

- 1 

X 

_ Ij 

(39) 


or 


V=^  A  fe 


(40) 


linking  the  Fourier  Transform  of  acceleration  V  with  the  force  vector  fg  of 
the  moving  forces  in  the  time  domain.  Wg  is  the  sub-matrix  of  W.  If  N  = 
fg  can  be  found  by  solving  the  Mh  order  linear  equations.  If  N>  or  more 
than  one  accelerations  are  measured,  the  least  squares  method  can  be  used  to 
find  the  time  history  of  the  moving  force  f(t). 

Equation  (40)  can  be  rewritten  as  follows 


(41) 


relating  the  accelerations  and  force  vectors  in  the  time  domain.  Also  if  N  - 
Nb,  fe  can  be  found  by  solving  the  Nth  order  linear  equation.  If  N  >  Nb  or 
more  than  one  acceleration  are  measured,  the  least  squares  method  can  be 
used  to  find  the  time  history  of  the  moving  force  f(t). 


If  only  Nc  {Nc  ^  N)  response  data  points  of  the  beam  are  used,  the 
equations  for  these  data  points  in  Equation  (41)  are  extracted,  and  described  as 


V,  =(w*)"  A 

C  V  B  /  A/x/V 


c 


B 

Nr  X  N 


W3 

/Vx;V, 


yV„x! 


(42) 


1093 


In  usual  cases  Nc  >  Ng,  so  the  least  squares  method  is  used  to  find  the  time 
history  of  the  moving  force  f(t).  More  than  one  acceleration  measurements  at 
different  locations  can  be  used  together  to  identify  a  single  moving  force  for 
higher  accuracy. 


B.  Identification  from  Bending  Moments  and  Accelerations 


Similarly,  the  relationships  between  bending  moment  m  (and  M )  and 
the  moving  force  f  can  be  described  as  follows, 

M  =  ^  B  W  is  (43) 

/Vxl  N  N^Nn 


(44) 

(45) 


The  force  vector  fg  can  be  obtained  from  the  above  three  sets  of 
equations.  Furthermore,  these  equations  can  be  combined  with  Equations  (40), 
(41)  and  (42)  to  construct  overdetermined  equations  before  the  equations  are 
scaled.  Two  forces  identification  are  developed  using  the  similar  procedure  as 
that  for  the  Time  Domain  Method. 


COMPARATIVE  STUDY 

The  first  moving  force  identification  method  is  implemented  in  a 
computer  program  using  FORTRAN,  while  the  other  three  methods  are 
implemented  under  the  environment  of  a  high  performance  numerical 
computation  and  visualization  software.  The  predictive  analysis  using  beam- 
elements  model  is  used  to  generate  the  theoretical  bridge  responses  and  the 
four  moving  force  identification  methods  then  use  these  responses  to  recover 
the  original  dynamic  loads.  In  this  study,  if  at  least  80%  of  the  identified 
forces  at  any  instant  of  any  load  lie  within  ±  1 0%  of  the  original  input  force, 
the  method  is  considered  acceptable.  It  is  found  that  all  the  four  methods  can 
give  acceptable  results. 

It  is  decided  to  carry  out  a  preliminary  comparative  study  on  the  four 
methods  in  order  to  study  the  merits  and  limitations  of  each  method  so  as  to 
consider  the  future  development  of  each  method  and  devise  a  plan  to  develop  a 


1094 


moving  force  identification  system  which  can  make  use  of  the  benefits  of  all 
the  four  methods. 


A.  Maximum  Number  of  Forces 


This  is  to  examine  the  maximum  number  of  axle  loads  that  can  be 
identified  by  each  method.  Theoretically,  provided  that  sufficient  number  of 
nodal  sensors  are  installed,  IMI  and  IMII  can  be  used  to  identify  as  many  loads 
as  the  system  allows.  Basically,  the  number  of  axle  loads  cannot  be  larger  than 
the  number  of  nodal  sensors.  Regarding  TDM  and  FDM,  as  the  formulation  of 
the  governing  equation  is  derived  for  two  moving  forces,  the  maximum 
number  of  axle  loads  that  can  be  identified  is  two. 

B.  Minimum  Number  of  Sensors 


Based  on  a  study  of  common  axle  spacings  of  vehicles  currently 
operating  on  Australian  roads,  and  the  cases  with  zero  nodal  responses, 
0‘ Connor  and  Chan  [6]  state  the  relationships  of  the  minimum  number  of 
sensors  used  for  IMI  and  the  span  length  of  a  bridge  as  follows: 


Using  bending  moment,  for  span  length  L  >  4.8m, 

Min.  number  of  nodal  moments  required  =  int{ 
Using  displacements,  for  span  length  L  >  13.8m, 

Min.  number  of  nodal  displacement  required  =  INj{ 
and  for  span  length  L  <13.8m, 

Min.  number  of  nodal  displacement  required 


L  -  4.8' 
1.7 

L-UX 


3.7 


+  4 


+  6 


(46) 

(47) 

(48) 


For  IMII,  it  is  found  that  the  number  of  sensors  required  are  generally  less 
than  that  for  IMI.  Regarding  TDM  and  FDM,  the  programs  are  not  as  flexible 
as  that  for  IMI  and  IMII  and  it  is  not  easy  to  change  the  number  of  sensors. 
Meanwhile  the  sensors  are  fixed  to  be  at  1/4,  1/2  and  3/4  of  the  span. 

C.  Sensitivity  towards  Noise 


In  general,  all  the  four  methods  can  compute  the  identified  forces  exactly 
the  same  as  those  given  to  the  predictive  analysis  to  generate  the 
corresponding  responses.  It  is  decided  to  add  white  noise  to  the  calculated 
responses  to  simulate  polluted  measurements  and  to  check  their  sensitivity 
towards  noise.  The  polluted  measurements  are  generated  by  the  following 


1095 


equations: 


m  =  n>o.nkui«.«]  x  N„, 


(49) 


where  Ep  is  a  specified  error  level;  is  a  standard  normal  distribution 
vector  (with  zero  mean  value  and  unity  standard  deviation). 

Several  cases  are  studied  using  Ep  =1%,  3%,  5%  and  10%.  It  is  found 
that  when  using  bending  moments  for  IMI  and  IMII,  and  if  Ep  is  less  than  3%, 
acceptable  results  can  be  obtained.  For  noise  which  is  greater  than  3%,  a 
smoothing  scheme  should  be  adopted  to  smooth  the  simulated  data. 
Acceptable  results  cannot  be  obtained  for  Ep  >  10%.  Besides,  both  IMI  and 
IMII  cannot  give  acceptable  results  when  using  displacements. 

Both  TDM  and  FDM  cannot  give  acceptable  results  when  using 
displacements  only,  accelerations  only  or  bending  moments  only.  In  general 
TDM  and  FDM  are  less  sensitive  to  noise  when  comparing  to  IMI  and  IMII. 
They  can  give  acceptable  results  for  Ep  up  to  5  %  without  any  smoothing  of 
the  polluted  simulated  data. 

D.  Computation  Time 

In  general,  the  computer  program  for  IMI  only  takes  few  seconds  to 
identify  moving  forces.  In  order  to  compare  the  computation  time,  IMI  is 
implemented  in  the  same  environment  as  the  other  three  methods.  It  is  found 
that  IMI  and  IMII  take  about  2-3  minutes  to  give  the  identified  forces  for  a 
case  of  two  axle  loads  using  a  80486  computer.  However,  under  the  same 
working  conditions,  TDM  and  FDM  almost  take  a  whole  day  for  any  one  of 
them  to  identify  two  moving  forces.  It  is  due  to  the  fact  that  both  of  them 
require  to  set  up  an  huge  parametric  matrix. 


CONCLUSIONS 

Four  methods  are  developed  to  identify  moving  time-varying  force  and 
they  all  can  produce  acceptable  results.  From  a  preliminary  comparative  study 
of  the  methods,  it  is  found  that  IMI  and  IMII  have  a  wider  applicability  as  the 
locations  of  sensors  are  not  fixed  and  it  can  identify  more  than  two  moving 
forces.  However,  TDM  and  FDM  are  less  sensitive  to  noise  and  require  less 
number  of  sensors.  It  is  decided  to  further  improve  the  four  methods  and  then 
a  more  detailed  and  systematic  comparison  can  be  carried  out  afterwards.  The 
possible  development  of  the  methods  are  described  as  follows. 


1096 


Both  the  IMI  and  IMII  are  developed  to  work  with  one  kind  of  responses, 
e.g.  either  displacements  or  bending  moments.  It  is  suggested  to  modify  the 
programs  to  use  mixed  input  parameter,  e.g.  use  bending  moments  as  well  as 
accelerations  as  that  for  TDM  or  FDM.  Regarding  the  TDM  and  FDM,  as  the 
basic  formulations  are  based  on  two-axle  moving  forces,  so  it  is  necessary  to 
modify  the  governing  equations  for  multi-axle.  In  addition,  the  computation 
time  for  TDM  or  FDM  under  the  environment  of  the  high  performance 
numerical  computation  and  visualization  software  used  is  unbearable.  It  is 
expected  that  the  time  will  be  significantly  reduced  if  the  methods  are 
implemented  in  programs  using  standard  programming  languages  like 
FORTRAN  90  or  C.  Then  the  four  methods  can  be  combined  together  and 
merged  into  a  Moving  Force  Identification  System  (MFIS)  so  that  it  can 
automatically  select  the  best  solution  routines  for  the  identification. 


ACKNOWLEDGMENT 

The  present  project  is  funded  by  the  Hong  Kong  Research  Grants  Council. 


REFERENCES 

1.  Davis,  P.  and  Sommerville,  F.,  Low-Cost  Axle  Load  Determination, 
Proceedings,  13th  ARRB  &  5th  REAAA  Combined  Conference,  1986, 
Part  6,  p  142-149. 

2.  Peters,  R.J.,  AXWAY  -  a  System  to  Obtain  Vehicle  Axle  Weights, 
Proceedings,  12th  ARRB  Conference,  1984,  12  (2),  p  10-18. 

3.  Peters,  R.J.,  CULWAY  -  an  Unmanned  and  Undetectable  Highway  Speed 
Vehicle  Weighing  System,  Proceedings,  13th  ARRB  &  5th  REAAA 
Combined  Conference,  1986,  Part  6,  p  70-83. 

4.  Cebon,  D.  Assessment  of  the  Dynamic  Wheel  Forces  Generated  by 
Heavy  Vehicle  Road  Vehicles.  Symposium  on  Heavy  Vehicle  Suspension 
Characteristics,  ARRB,  1987. 

5.  Cantineni,  R.,  Dynamic  Behaviour  of  Highway  Bridges  Under  The 
Passage  of  Heavy  Vehicles.  Swiss  Federal  Laboratories  for  Materials 
Testing  and  Research  (EMPA)  Report  No.  220,  1992,  240p. 

6.  O'Connor,  C.  and  Chan,  T.H.T.,  Dynamic  Wheel  Loads  from  Bridge 
Strains.  Structural  Engineering  ASCE,  1 14  (STS),  1988,  p.  1703- 1723. 

7.  Briggs,  J.C.  and  Tse,  M.K.  Impact  Force  Identification  using  Extracted 
Modal  Parameters  and  Pattern  Matching,  International  Journal  of  Impact 
Engineering,  1992,  Vol.  12,  p361-372. 

8.  Bendat,  J.S.  and  Piersol,  J.S.,  Engineering  Application  of  Correlation  and 
Spectral  Analysis.  John  Wiley  &  Sons,  Inc.  Second  Edition,  1993. 


1097 


1098 


ESTIMATING  THE  BEHAVIOUR  OF  A 
NONLINEAR  EXPERIMENTAL  MULTI  DEGREE 
OF  FREEDOM  SYSTEM  USING  A  FORCE 
APPROPRIATION  APPROACH 


P.A.  Atkins  J.R. Wright 

Dynamics  and  Control  Research  Group 
School  of  Engineering,  Simon  Building,  University  of  Manchester, 
Oxford  Road,  Manchester.  M13  9PL 


ABSTRACT 

The  identification  of  nonlinear  multi  degree  of  freedom  systems  involves  a 
significant  number  of  nonlinear  cross  coupling  terms,  whether  the  identifi¬ 
cation  is  carried  out  in  spatial  or  modal  domains.  One  possible  approach 
to  reducing  the  order  of  each  identification  required  is  to  use  a  suitable 
pattern  of  forces  to  drive  any  mode  of  interest.  For  a  linear  system,  the 
force  pattern  required  to  drive  a  single  mode  is  derived  using  a  Force  Ap¬ 
propriation  method.  This  paper  presents  a  method  for  determining  the 
force  pattern  necessary  to  drive  a  mode  of  interest  of  a  nonlinear  system 
into  the  nonlinear  region  whilst  the  response  is  controlled  to  remain  in  pro¬ 
portion  to  the  linear  mode  shape.  Such  an  approach  then  allows  the  direct 
nonlinear  modal  terms  for  that  mode  to  be  identified  using  the  Restoring 
Force  method.  The  method  for  determining  the  relevant  force  patterns  is 
discussed.  The  implementation  of  the  method  for  experimental  systems  is 
considered  and  experimental  results  from  a  two  degree  of  freedom  ’bench¬ 
mark  structure’  are  presented. 


INTRODUCTION 

Force  Appropriation  [1]  is  used  in  the  analysis  of  linear  systems  to  de¬ 
termine  the  force  patterns  which  will  induce  single  mode  behaviour  when 
applied  at  the  relevant  natural  frequency.  This  technique  is  used  in  the 
aerospace  industry  during  Ground  Vibration  Tests:  each  normal  mode  of 


1099 


a  structure  is  excited  using  the  derived  force  pattern  and  thus  identified 
in  isolation.  Current  practice,  when  the  presence  of  nonlinearity  is  sus¬ 
pected,  is  to  increase  input  force  levels  and  monitor  the  variation  of  tuned 
frequencies.  Some  information  about  the  type  of  nonlinearity  present  may 
be  found,  but  no  analytical  model  can  be  derived.  Thus  predictions  for 
behaviour  at  higher  levels  of  excitation  axe  not  possible. 

A  number  of  techniques  for  identifying  nonlinearity,  for  example  the  Restor¬ 
ing  Force  method  [2],  have  been  demonstrated  on  systems  with  low  num¬ 
bers  of  degrees  of  freedom.  Unfortunately,  in  practice,  structures  have  a 
large  number  of  degrees  of  freedom,  often  with  a  high  modal  density.  A 
classical  Restoring  Force  approach  to  the  identification  of  such  systems 
could  involve  a  prohibitive  number  of  cross  coupling  terms.  The  ability  to 
treat  each  mode  separately,  by  eliminating  the  effects  of  the  cross  coupling 
terms,  would  thus.be  advantageous.  Subsequent  tests  could  then  evaluate 
the  cross  coupling  terms. 

For  these  reasons  it  would  be  useful  to  extend  Force  Appropriation  to  the 
identification  of  nonlinear  systems.  An  approach  has  been  developed  [6] 
that  allows  an  input  force  pattern  to  be  derived  that  will  result  in  a  non¬ 
linear  response  in  the  linear  mode  shape  of  interest.  This  force  pattern  is 
derived  using  an  optimisation  approach.  The  mode  of  interest  can  then  be 
identified  using  a  single  degree  of  freedom  nonlinear  identification  method. 
In  this  work  the  Restoring  Force  method  is  used  to  examine  the  nonlinear 
response  of  a  particular  linear  mode  and  an  application  of  this  approach 
to  a  two  degree  of  freedom  experimental  system  is  presented. 


THEORY 


The  theoretical  approach  is  demonstrated  for  the  two  degree  of  freedom 
system  with  spring  grounded  nonlinearity  shown  in  figure  1.  The  equations 
of  motion  for  this  system  in  physical  space  are: 


m  0 
0  m 


+ 


+ 


(1  +  a)c 

—ac 

'±1 1 

—ac 

(1  +  d)c 

\ 

X2  J 

(l  -}-  Q^k 

—ak 

fill 

—ak 

(1  +  a)k 

l^x\ 

0 


where  is  the  cubic  stifEhess  coefficient  and  a  is  a  constant  that  allows  the 
frequency  spacing  of  the  natural  frequencies  to  be  varied.  These  equations 
can  be  transformed  to  linear  modal  space  using  the  transformation: 


{i}  =  [ij>]{u}  (2) 

where  [(j>]  is  the  modal  matrix  of  the  underlying  linear  system  and  the 
vector  {u}  defines  the  modal  displacements.  For  this  symmetrical  system 


1100 


the  modal  matrix  is 


1  1 
1  -1 


(3) 


The  equations  of  motion  transformed  to  linear  modal  space  using  the  nor¬ 
malised  modal  matrix  are: 


771  0  \  ill 

0  771  (112 


+ 


c  0 

0  (1  +  2g)c 

1^2/ 

■jb  0 

Ull 

0  (1  +  2a)k 

1^2] 

P{Ul-U2y/4:' 

-U2)V4 


where  {p}  is  the  modal  input  vector.  It  can  be  seen  from  the  above  equa¬ 
tions  that  the  cubic  nonlinearity  couples  the  modes  in  linear  mod^  space; 
in  fact  there  are  a  significant  number  of  terms  for  a  single  nonlinearity. 
The  proposed  method  aims  to  determine  the  force  pattern  that  will  reduce 
the  response  of  this  system  to  that  of  a  single  mode. 

It  was  shown  in  a  previous  paper  [3]  that  this  can  be  achieved  by  seeking  a 
force  vector  that  will  cause  motion  only  in  the  target  mode,  by  eliminating 
motion  in  the  coupled  mode.  In  practice,  physical  data  from  transducers 
are  available.  Any  subsequent  transformations  would  be  time  consuming. 
It  is  shown  below  that  causing  motion  in  one  mode  to  be  zero  is  equiva¬ 
lent  to  forcing  motion  in  a  linear  mode  shape,  mode  one  in  this  example. 
Consider  the  coordinate  transformation  {a;}  =  [<^]{ii}  or  more  explicitly 
for  the  two  degree  of  freedom  system  in  Figure  1: 


Thus 


1 

2 


‘l  -1 
1  1 


(5) 

(6) 


and  enforcing  the  first  mode  shape  (1, 1)  in  physical  space  should  give  a 
second  modal  displacement  of  zero. 


1 

2 


1  -1 
1  1 


1 

1 


(7) 


So  if  an  excitation  is  applied  which  causes  the  nonlinear  system  to  vibrate 
in  its  first  linear  mode  shape,  the  response  will  be  composed  only  of  ui  and 
the  influence  of  the  coupled  mode,  U2,  will  then  have  been  eliminated. 

The  method  must  therefore  derive  a  force  pattern  which  will  cause  the 
system  to  vibrate  in  one  of  its  linear  mode  shapes.  It  has  been  shown 


1101 


in  a  previous  paper  [3]  that  if  the  response  contains  harmonics  then  the 
force  pattern  must  also  contain  harmonics  in  order  to  control  the  harmonic 
content  of  the  response.  In  theory,  the  responses  will  be  an  infinite  series 
of  harmonics,  but  this  series  is  truncated  in  this  case  of  a  cubic  stiffness 
nonlinearity  to  include  only  the  fundamental  and  third  harmonic  terms. 

The  physical  input  forces  will  thus  be  of  the  form: 

fi(t)  =  Fii  cos(a;ea;t  4-  ^ii)  4-  F13  cos(a;ea;t  +  (^13)  (8) 

f2{t)  =  F21  C0s(a;ea;t  +  (j>2l)  4*  F23  COS{uJext  +  fe)  (9) 

where  is  the  excitation  frequency.  Parameters  for  these  force  patterns 
may  then  be  chosen  such  that  only  mode  one  is  excited. 


OPTIMISATION  APPROACH 

In  general,  no  a  priori  model  of  the  system  exists  so  an  optimisation  routine 
is  used  to  determine  the  force  pattern  parameters  required  to  maximise  the 
contribution  of  the  mode  of  interest.  The  objective  function,  the  quantity 
that  the  optimisation  routine  seeks  to  minimise,  must  be  representative 
of  the  deviation  of  the  response  from  the  target  linear  mode  shape.  The 
objective  function,  F,  that  was  chosen  in  this  case  was  based  on  the  vector 
norm  [4]  of  the  two  physical  responses,  Xi  and  X2,  and  is  shown  below: 


where  and  02  are  elements  of  the  mode  shape  vector  for  the  target 
mode.  This  summation  is  carried  out  over  one  cycle  of  the  fundamental 
response.  The  number  of  data  points  per  cycle  is  npts  and  Xki  the  kth 
response  at  the  itk  sample.  This  objective  function  allows  the  response 
to  contain  harmonics  and  can  be  extended  to  more  degrees  of  freedom  by 
choosing  a  reference  displacement  and  subtracting  further  displacements 
from  it.  The  Variable  Metric  optimisation  method  [5]  was  used  in  this 
work  as  it  has  been  found  to  produce  the  best  results  for  simulated  data. 
The  application  of  this  method  to  a  two  degree  of  freedom  system  such 
as  that  shown  in  figure  1  is  detailed  in  [6].  Optimised  force  patterns  are 
obtained  at  several  levels  of  input  amplitude.  These  force  patterns  are  then 
applied  and  the  Restoring  Force  method  is  used  to  curve  fit  the  resulting 
modal  displacement  and  velocity  time  histories  to  give  the  direct  linear  and 
nonlinear  coefficients  for  the  mode  of  interest. 


1102 


EXPERIMENTAL  IMPLEMENTATION 


The  simulated  application  of  this  method  assumed  that  certain  parame¬ 
ters  were  known.  In  order  to  carry  out  an  identification  of  an  experimental 
structure,  these  parameters  must  be  measured  or  calculated.  Some  pro¬ 
cessing  of  experimental  data  is  necessary  in  order  to  apply  the  Restoring 
Force  method.  The  restoring  force  of  a  system  can  be  expressed  for  a  single 
degree  of  freedom  system  as: 

h{x,  x)  =  f(t)  -  mx  (11) 

where  h{x,  x)  is  the  restoring  force  and  f{t)  the  input  force.  A  similar  ex¬ 
pression  applies  to  the  modal  restoring  force  for  an  isolated  mode.  Thus  the 
input  force,  acceleration,  velocity  and  displacement  must  be  calculated  at 
the  each  time  instant.  A  similar  expression  applies  to  the  modal  restoring 
force  for  an  isolated  mode.  In  the  experimentaJ  case  it  is  usual  to  measure 
acceleration  and  input  force;  the  remaining  two  states  must  therefore  be 
obtained  by  integration  of  the  acceleration  time  history.  Frequency  domain 
integration  [7]  was  used  for  this  purpose.  High  pass  filtering  was  used  to 
remove  any  low  frequency  noise  which  can  be  amplified  by  this  type  of 
integration.  Several  methods  have  been  suggested  for  estimating  modal 
mass,  but  in  this  study  a  method  developed  by  Worden  and  Tomlinson  [8] 
was  used.  An  estimate  for  the  modal  mass  is  obtained  and  then  an  error 
term  is  included  in  the  curve  fit  which  will  iteratively  yield  a  more  accu¬ 
rate  estimate  of  the  mass.  Generally  the  mass  value  will  converge  after  one 
iteration. 

The  objective  function  used  in  the  simulations  was  calculated  from  the 
displacement  time  histories.  In  the  experimental  case,  acceleration  was 
used  rather  than  displacement  as  it  was  considered  that  using  ’raw’  data 
would  be  quicker  and  give  less  opportunity  for  error.  In  the  simulated 
case,  the  system  parameters  were  known  a  priori  so  the  modal  matrix 
of  the  underlying  linear  system  could  be  calculated.  For  most  types  of 
nonlinearity  the  response  of  the  system  at  low  input  force  levels  will  be 
dominated  by  linear  terms.  Normal  mode  tuning  [1]  was  therefore  applied 
at  low  force  levels  to  yield  an  approximation  to  the  modal  matrix  of  the 
underlying  linear  system. 

A  quality  indicator  to  give  some  idea  of  the  effectiveness  of  the  optimisa¬ 
tion  performed  would  be  advantageous.  Results  corrupted  by  background 
noise,  for  example,  could  then  be  discarded.  A  perfect  optimisation  will 
occur  when  the  ratio  of  measured  accelerations  exactly  matches  the  mode 
shape  ratio  specified  for  the  mode  of  interest.  Thus  a  least  squares  fit  of 
the  sampled  accelerations  was  carried  out  over  a  cycle  of  the  fundamental 
frequency  and  the  percentage  error  of  the  measured  mode  shape  to  the 


1103 


required  mode  shape  was  calculated.  This  percentage  error  will  indicate 
whether  the  optimisation  has  been  successful. 

To  assess  the  accuracy  of  the  parameters  estimated  using  this  method, 
an  identification  was  carried  out  using  a  conventional  Restoring  Force  ap¬ 
proach  in  physical  space.  A  band  limited  random  excitation  was  used,  and 
the  physical  data  processed  and  curve  fitted.  The  physical  parameters  were 
then  transformed  to  modal  space.  The  direct  linear  and  nonlinear  param¬ 
eters  for  modes  one  and  two  are  shown  in  table  1.  It  should  be  noted  that 
although  this  conventional  Restoring  Force  approach  is  possible  for  this 
two  degree  of  freedom  system,  it  will  not  generally  be  possible  since  the 
number  of  terms  in  the  curve  fit  increases  dramatically  when  different  t3q)es 
of  nonlinearity  and  more  degrees  of  freedom  are  included.  It  is  carried  out 
in  this  case  as  a  means  of  validating  the  proposed  method. 


EXPERIMENTAL  SETUP 

The  rig  constructed  consisted  of  two  masses  on  thin  legs  connected  in  series 
by  a  linear  spring,  each  mass  being  driven  by  a  shaker.  A  cubic  nonlin¬ 
earity  was  introduced  between  the  first  mass  and  ground  using  a  clamped- 
clamped  beam  attached  at  the  centre  which  will  yield  a  cubic  stiffness  for 
large  deflections  [9],  A  schematic  diagram  of  the  rig  is  shown  in  figure  2. 
The  force  input  by  each  shaker  was  measured  using  a  force  gauge  and  the 
acceleration  of  each  mass  was  measured  using  an  accelerometer  in  the  po¬ 
sitions  also  shown  in  figure  2,  Acceleration  and  force  data  were  acquired 
using  a  multiple  channel  acquisition  system,  the  optimisation  routine  was 
carried  out  on  line. 


RESULTS 


Normal  mode  tuning  of  the  rig  gave  natural  frequencies  of  20.67  Hz  and 
24.27  Hz  and  a  modal  matrix  of: 


3.87  5.03 
5.52  -3.27 


(12) 


The  excitation  frequency  was  chosen  to  be  slightly  lower  than  the  natural 
frequency  of  the  mode  of  interest  in  order  to  avoid  the  problems  associated 
with  force  drop  out  which  are  worst  at  the  natural  frequency.  For  each 
mode  optimisation  was  performed  at  three  input  force  levels,  the  highest 
level  was  as  high  as  possible  so  as  to  excite  the  nonlinearity  strongly.  The 
optimisation  routine  was  carried  out  using  the  voltage  input  into  the  signal 
generator  as  the  variable.  The  force  input  into  the  structure  was  measured 


1104 


for  use  in  the  Restoring  Force  identification  but  was  not  used  in  the  opti¬ 
misation  as  it  is  not  directly  controllable.  The  details  of  the  optimisation 
for  each  force  level  are  presented  in  table  2.  The  optimised  forces  and  re¬ 
sulting  accelerations  for  mode  one  are  shown  in  figure  3  and  figure  4.  The 
acceleration  data  for  the  optimised  force  patterns  were  then  integrated  and 
the  modal  restoring  force  for  the  mode  of  interest  calculated.  The  initial 
estimate  of  modal  mass  for  the  calculation  of  the  modal  restoring  force  was 
tahen  from  a  previous  paper  [10]  in  which  the  rig  was  identified  using  a 
using  a  physical  parameter  identification  method.  The  mass  was  estimated 
in  this  paper  to  be  2.62  kg,  this  physical  mass  will  then  be  equal  to  the 
modal  mass  since  the  modal  matrix  was  normalised  to  be  orthonormal. 
The  restoring  force  data  was  then  transformed  to  modal  space.  The  modal 
restoring  force  surface  obtained  using  optimised  force  inputs  for  mode  one 
is  shown  in  figure  5  and  a  stiffness  section  through  this  surface  is  shown  in 
figure  6.  The  restoring  force  time  histories  were  then  curve  fitted  against 
modal  velocity  and  displacement. 

The  estimated  parameters  for  mode  one  axe  shown  in  table  3.  It  can  be 
seen  that  they  do  not  compare  very  well  with  those  estimates  obtained 
using  the  band  limited  random  excitation.  It  was  suspected  that  at  lower 
excitation  levels  the  estimates  were  being  distorted  by  linear  dependence 
[11].  Linear  dependence  is  a  problem  which  occurs  when  curve  fitting  a 
harmonic  response  from  a  linear  system;  the  equations  of  motion  may  be 
identically  satisfied  by  mass  and  stiffness  terms  modified  by  an  arbitrary 
constant.  This  condition  is  avoided  by  the  harmonic  terms  introduced  into 
the  response  by  nonlinearity.  The  curve  fit  was  thus  repeated  using  only  the 
data  obtained  from  the  highest  level  of  excitation;  the  estimates  obtained 
are  shown  in  table  4.  It  can  be  seen  that  the  linear  parameter  estimates 
now  agree  well  with  the  band  limited  random  results.  The  estimates  for 
the  cubic  stiffness  coefficient  do  not  appear  to  agree  so  well.  The  standard 
deviation  on  the  cubic  stiffness  derived  from  the  band  limited  random 
excitation  is  approximately  a  third  of  the  value  of  the  parameter  itself.  The 
uncertainty  on  this  parameter  occurs  because  the  nonlinearity  is  not  very 
strongly  excited  by  this  type  of  excitation.  A  stiffness  section  through  the 
restoring  force  surface,  figure  7,  shows  little  evidence  of  a  cubic  stiffness 
component.  If  a  higher  level  of  excitation  were  possible  then  a  better 
estimate  may  be  achieved. 

The  identification  w^  repeated  for  mode  two.  The  restoring  force  surface 
obtained  and  a  stiffness  section  through  it  are  shown  in  figure  8  and  figure  9. 
It  can  be  seen  from  the  stiffness  section  that  the  nonlinearity  is  not  very 
strongly  excited.  The  estimated  direct  modal  parameters  are  shown  in 
table  5.  It  can  be  seen  that  these  results  agree  quite  well  with  those 
obtained  using  band  limited  random  excitation.  It  is  considered  that  the 
discrepancy  between  the  two  sets  of  results,  in  particular  the  mass  and 


1105 


stiifiiess  estimates,  is  again  due  to  linear  dependence. 


CONCLUSIONS 

An  extension  of  the  force  appropriation  method  has  been  proposed  for 
nonlinear  systems.  In  this  method,  an  optimisation  routine  is  used  to 
determine  the  force  patterns  which  will  excite  a  single  mode  nonlinear 
response  .  The  direct  linear  and  nonlinear  modal  parameters  can  then  be 
estimated  from  a  curve  fit  of  the  modal  restoring  force  surface.  The  method 
was  applied  to  an  experimental  two  degree  of  freedom  system  whose  modes 
were  coupled  in  linear  modal  space  by  a  spring  grounded  nonlinearity. 
A  conventional  restoring  force  identification  was  performed  using  a  band 
limited  random  signal  for  comparison.  The  parameters  estimated  from  the 
single  mode  responses  were  found  to  agree  quite  well  with  those  from  the 
band  limited  random  tests. 


REFERENCES 


1.  Holmes  P.,  Advanced  Applications  of  Normal  Mode  Testing,  PhD 
Thesis,  University  of  Manchester  1996. 

2.  Hadid  M.A.  and  Wright  J.R.,  Application  of  Force  State  Mapping 
to  the  Identification  of  Nonlinear  Systems  Mechanical  Systems  and 
Signal  Processing,  1990,  4(6),  463-482 

3.  Atkins  P.A.,  Wright  J.R.,  Worden  K.,  Manson  G.M.  and  Tomlinson 
G.R.,  Dimensional  Reduction  for  Multi  Degree  of  Freedom  Nonlinear 
Systems,  International  Conference  on  Identification  in  Engineering 
Systems  1996,  712-721 

4.  Kreyszig  E.,  Advanced  Engineering  Mathematics,  Wiley,  6th  Edition 

5.  Press  W.H.,  Teukolsky  S.A.,  Vettering  W.T.  and  Flannery  B.P.,  Nu¬ 
merical  Recipes  in  Fortran  Cambridge  University  Press,  6th  Edition 

6.  Atkins  P.A.  and  Wright  J.R.,  An  Extension  of  Force  Appropriation 
for  Nonlinear  Systems  Noise  and  Vibration  Engineering,  Proceedings 
of  ISMA21(2),  915-926,  1996 

7.  Worden  K.,  Data  Processing  and  Experiment  Design  for  the  Restor¬ 
ing  Force  Method,  Part  I:  Integration  and  Differentiation  of  Measured 
Time  Data,  Mechanical  Systems  and  Signal  Processing,  4(4)  295-319, 
1990 


1106 


8.  Ajjan  Al-Hadid  M.,  Identification  of  Nonlinear  Dynamic  Systems  us¬ 
ing  the  Force  State  Mapping  Technique,  PhD  Thesis,  Queen  Mary 
College,  University  of  London,  1989 

9.  Storer  D.M.,  Dynamic  Analysis  of  Nonlinear  Structures  Using  Higher 
Order  Frequency  Response  Function,  PhD  Thesis,  University  of  Manch¬ 
ester,  1991 

10.  Atkins  P.  and  Worden  K.,  Identification  of  a  Multi  Degree  of  Freedom 
Nonlinear  System,  Proc.  of  IMAC  XV,  1997 


ACKNOWLEDGEMENTS 

This  work  was  supported  by  E.P.S.R.C.  under  research  grant  number 
GR/J48238  at  the  University  of  Manchester 


Modal  parameter 

Mode  one 

Mode  two 

k  (N/m) 
c  (Nm/s) 

/3(N/m^) 
m  (kg) 

4.87  X  10^ 
10.11 

3.83  X  10® 
2.60 

6.49  X  10^ 
9.49 

8.90  X  10* 
3.06 

Table  1:  Direct  modal  parameters  estimated  from  curve  fit  of  band  limited 
random  data 


Low  forcing 

Medium  forcing 

High  forcing 

Fii  (Volts) 

1.0 

2.0 

3.0 

F21  (Volts) 

1.0 

2.0 

3.0 

u)^  (Hz) 

20.0 

20.0 

20.0 

initial  mode  shape  ratio 

0.72 

0.68 

0.65 

final  mode  shape  ratio 

0.70 

0.70 

0.70 

target  mode  shape  ratio 

0.70 

0.70 

0.70 

percentage  error 

0.14 

0.03 

0.71 

Table  ,2:  Details  of  optimisation  for  mode  one 


Table  3:  Direct  modal  parameters  estimated  from  optimised  responses 


Model  parameter  Estimated  parameter 
k  (N/m)  4.57  X  10^ 

c  (Nm/s)  8.62 

/?(N/m3)  6.81  X  10® 

m  (kg) _ _ 2^75 _ 

Table  4:  Direct  modal  parameters  estimated  using  high  force  level  only 


Model  parameter 

Estimated  parameter 

k  (N/m) 
c  (Nm/s) 

m  (kg) 

5.37  X  10^ 

10.16 

2.45  X  10® 

2.29 

Table  5:  Direct  modal  parameters  estimated  for  mode  two 


Figure  1:  Two  degree  of  freedom  system 


1108 


0  s  0.40008 


Figure  3:  Optimised  forces  for  mode  one  at  a  high  force  level 


Figure  4:  Accelerations  responses  to  optimised  forces 


Figure  5:  Modal  restoring  force  surface  for  mode  one 


Force  (N) 


Figure  7:  Stiffness  section  through  modal  restoring  force  derived  from  ran¬ 
dom  excitation  for  mode  one 


1111 


Figure  8:  Modal  restoring  force  surface  for  mode  two 


Figure  9:  Stiffness  section  through  modal  restoring  force  for  mode  two 


1112 


POWER  FLOW  TECHNIQUES  II 


THE  OPTIMAL  DESIGN  OF  NEAR-PERIODIC  STRUCTURES  TO 
MINIMISE  NOISE  AND  VIBRATION  TRANSMISSION 


R.S.  Langley,  N.S.  Bardell,  and  P.M.  Loasby 
Department  of  Aeronautics  and  Astronautics 
University  of  Southampton 
Southampton  SO  17  IBJ,  UK 

1.  INTRODUCTION 

An  engineering  structure  is  said  to  be  of  "periodic"  construction  if  a  basic 
structural  unit  is  repeated  in  a  regular  pattern  .  A  beam  which  rests  on 
regularly  spaced  supports  is  one  example  of  a  one-dimensional  periodic 
structure,  while  an  orthogonally  stiffened  cylinder  is  an  example  of  a  two- 
dimensional  periodic  structure.  It  has  long  been  known  that  perfectly  periodic 
structures  have  very  distinctive  vibration  properties,  in  the  sense  that  "pass 
bands"  and  "stop  bands"  arise:  these  are  frequency  bands  over  which  elastic 
wave  motion  respectively  can  and  cannot  propagate  through  the  structure  [1,2]. 
If  the  excitation  frequency  lies  within  a  stop  band  then  the  structural  response 
tends  to  be  localised  to  the  immediate  vicinity  of  the  excitation  source. 
Conversely,  if  the  excitation  frequency  lies  within  a  pass  band  then  strong 
vibration  transmission  can  occur,  and  it  is  generally  the  case  that  the  resonant 
frequencies  of  the  structure  lie  within  the  pass  bands. 

Much  recent  work  has  been  performed  concerning  the  effect  of  random 
disorder  on  a  nominally  periodic  structure  (see  for  example  [3-5]).  It  has  been 
found  that  disorder  can  lead  to  localisation  of  the  response  even  for  excitation 
which  lies  within  a  pass  band,  and  this  reduces  the  propensity  of  the  structure 
to  transmit  vibration.  This  raises  the  possibility  of  designing  disorder  into  a 
structure  in  order  to  reduce  vibration  transmission,  and  this  possibility  was 
briefly  investigated  in  reference  [6]  for  a  one-dimensional  periodic  waveguide 
which  was  embedded  in  an  otherwise  infinite  homogeneous  system.  The 
present  work  extends  the  work  reported  in  reference  [6]  to  the  case  of  a  finite 
near-periodic  beam  system,  which  more  closely  resembles  the  type  of 
optimisation  problem  which  is.  likely  to  occur  in  engineering  practice.  The 
beam  is  taken  to  have  N  bays,  and  the  design  parameters  are  taken  to  be  the 
individual  bay  lengths.  Both  single  frequency  and  band-limited  excitation  are 
considered,  and  two  objective  functions  are  investigated:  (i)  the  response  in  a 
bay  which  is  distant  from  the  applied  loading  (minimisation  of  vibration 
transmission),  and  (ii)  the  maximum  response  in  the  structure  (minimisation 
of  maximum  stress  levels).  In  each  case  the  optimal  configuration  is  found  by 
employing  a  quasi-Newton  algorithm,  and  the  physical  features  of  the  resulting 
design  are  discussed  in  order  to  suggest  general  design  guidelines. 


1113 


2.  ANALYTICAL  MODEL  OF  THE  NEAR-PERIODIC  BEAM 


2.1  Calculation  of  the  Forced  Response 

A  schematic  of  an  N  bay  near-periodic  beam  structure  is  shown  in  Figure  1. 
The  structure  is  subjected  to  dynamic  loading,  and  the  aim  of  the  present  work 
is  to  find  the  optimal  design  which  will  minimise  a  prescribed  measure  of  the 
vibration  response.  No  matter  what  type  of  optimisation  algorithm  is 
employed,  this  type  of  study  requires  repeated  computation  of  the  system 
dynamic  response  as  the  design  parameters  are  varied,  and  it  is  therefore 
important  to  employ  an  efficient  analysis  procedure.  In  the  present  work  the 
h-p  version  of  the  finite  element  method  (FEM)  is  employed:  with  this 
approach  the  structure  is  modelled  as  an  assembly  of  elements  which  have 
both  nodal  and  internal  degrees  of  freedom.  Each  element  has  two  nodes  and 
the  nodal  degrees  of  freedom  consist  of  the  beam  displacement  and  slope;  the 
internal  degrees  of  freedom  are  generalised  coordinates  which  are  associated 
with  a  hierarchy  of  shape  functions  which  contribute  only  to  the  internal 
displacement  field  of  the  element.  The  internal  shape  functions  used  here  are 
the  K-orthogonal  Legendre  polynomials  of  order  four  onwards  -  full  details  of 
the  present  modelling  approach  are  given  in  reference  [7]. 

For  harmonic  excitation  of  frequency  co  the  equations  of  motion  of  the 
complete  beam  structure  can  be  written  in  the  form 

where  M  and  K  are  the  global  mass  and  stiffness  matrices  (assembled  from  the 
individual  element  matrices  taking  into  account  the  presence  of  any  mass  or 
spring  elements  and  allowing  for  constraints),  q  contains  the  system 
generalized  coordinates,  F  is  the  generalized  force  vector,  and  r|  is  the  loss 
factor,  which  in  the  present  study  is  taken  to  be  uniform  throughout  the 
structure. 

Equation  (1)  can  readily  be  solved  to  yield  the  system  response  q.  In  the 
present  work  it  is  convenient  to  use  the  time  averaged  kinetic  and  strain 
energies  of  each  of  the  N  bays  as  a  measure  of  the  response  -  for  the  nth  bay 
these  quantities  can  be  written  as  and  say,  where 

r,=(£0V4)9;>„?„.  (2.3) 

Here  and  K„  are  the  mass  and  stiffness  matrices  of  the  nth  bay,  and  q„  is 
the  vector  of  generalized  coordinates  for  this  bay. 

Many  of  the  physical  features  of  the  forced  response  of  a  near-periodic 
structure  can  be  explained  in  terms  of  the  free  vibration  behaviour  of  the 


1114 


associated  perfectly  periodic  structure.  The  following  section  outlines  how  the 
present  finite  element  modelling  approach  can  be  used  to  study  the  pass  bands 
and  stop  bands  exhibited  by  a  perfect  periodic  structure. 

2.2  Periodic  Structure  Analysis 

The  finite  element  method  described  in  section  2.1  can  be  applied  to  a  single 
bay  of  a  perfectly  periodic  structure  to  yield  an  equation  of  motion  in  the  form 

Dq=F,  D=-coW--(l+/r|)ii:,  (^’5) 


where  the  matrix  D  is  referred  to  as  the  dynamic  stiffness  matrix.  In  order  to 
study  wave  motion  through  the  periodic  system  it  is  convenient  to  partition  D, 
q  and  F  as  follows 


r 

Du 

ft: 

D= 

D„ 

D,r 

.  F= 

0 

Dr, 

F„ 

V  V 

(6-8) 


where  L  relates  to  the  coordinates  at  the  left  most  node,  R  relates  to  those  at 
the  right  most  node,  and  /  relates  to  the  remaining  "internal"  coordinates. 
Equations  (4-8)  can  be  used  to  derive  the  following  transfer  matrix  relation 
between  the  displacements  and  forces  at  the  left  and  right  hand  nodes 


V  / 


V  / 


(9,10) 

Equation  (9)  can  now  be  used  to  analyze  wave  motion  through  the  periodic 
system:  such  motion  is  governed  by  Bloch’s  Theorem,  which  states  that 
i^L  F i)=exp{-ie-b){qii  -F^)  where  8  and  5  are  known  respectively  as  the  phase 
and  attenuation  constants.  A  pass  band  is  defined  as  a  frequency  band  over 
which  6=0,  so  that  wave  motion  can  propagate  down  the  structure  without 
attenuation.  It  follows  from  equation  (9)  that 


(T-Ie 


-ie-6' 


(  \ 

0 

F, 

io 

V 

V  y 

(11) 


SO  that  8  and  6  can  be  computed  from  the  eigenvalues  of  T,  thus  enabling  the 
pass  bands  and  stop  bands  to  be  identified. 


1115 


2.3  Optimisation  Procedure 

Equations  (l)-(3)  enable  the  forced  response  of  the  system  to  be  calculated  for 
any  prescribed  set  of  system  properties.  The  aim  of  the  present  analysis  is  to 
compute  the  optimal  set  of  system  properties  for  a  prescribed  design  objective, 
and  in  order  to  achieve  this  equations  (l)-(3)  are  evaluated  repeatedly  as  part 
of  an  optimisation  algorithm.  As  an  example,  it  might  be  required  to  minimise 
the  kinetic  energy  of  bay  N  by  changing  the  various  bay  lengths.  In  this  case 
equations  (l)-(3)  provide  the  route  via  which  the  objective  function  (the  kinetic 
energy  in  bay  N)  is  related  to  the  design  parameters  (the  bay  lengths),  and  the 
optimisation  algorithm  must  adjust  the  design  parameters  so  as  to  minimise  the 
objective  function.  The  optimisation  process  has  been  performed  here  by  using 
the  NAg  library  routine  E04JAF  [8],  which  employs  a  quasi-Newton  algorithm. 
This  type  of  algorithm  locates  a  minimum  in  the  objective  function,  although 
there  is  no  indication  whether  this  minimum  is  the  global  minimum  or  a  less 
optimal  local  minimum.  The  probability  of  locating  the  global  minimum  can 
be  increased  significantly  by  repeated  application  of  the  NAg  routine  using 
random  starts,  i.e.  random  initial  values  of  the  design  parameters.  Numerical 
investigations  have  led  to  the  use  of  30  random  starts  in  the  present  work. 

3.  NUMERICAL  RESULTS 

3.1  The  System  Considered 

The  foregoing  analysis  has  been  applied  to  a  beam  of  flexural  rigidity  El,  mass 
per  unit  length  m,  and  loss  factor  r|=0.015,  which  rests  on  A+1  simple 
supports,  thus  giving  an  A’-bay  near-periodic  system.  The  design  parameters 
are  taken  to  be  the  bay  lengths  (i.e.  the  separation  of  the  simple  supports),  and 
the  design  is  constrained  so  that  the  length  of  any  bay  lies  within  the  range 
0.9L^,<1.1L,  where  L,  is  a  reference  length.  A  non-dimensional  frequency 
Q  is  introduced  such  that  0.=(oL,N{m/Er),  and  the  non-dimensional  kinetic  and 
strain  energies  of  a  bay  are  defined  as  T,'-T,^(EI/L^^\F\^)  and 
U,'~U„iEI/L,^\F\'^)  where  F  is  the  applied  point  load.  As  discussed  in  the 
following  subsections,  two  objective  functions  are  considered  corresponding 
to  minimum  vibration  transmission  and  minimum  overall  response.  In  all 
cases  the  excitation  consists  of  a  point  load  applied  to  the  first  bay  and  the 
response  is  averaged  over  1 1  equally  spaced  point  load  locations  within  the 
bay.  For  reference,  the  propagation  constants  for  a  periodic  system  in  which 
all  the  bay  lengths  are  equal  to  L,  are  shown  in  Figure  2  -  the  present  study  is 
focused  on  excitation  frequencies  which  lie  in  the  range  23<f2<61,  which 
covers  the  second  stop  band  and  the  second  pass  band  of  the  periodic  system. 

3.2  Design  for  Minimum  Vibration  Transmission 

In  this  case  the  objective  function  is  taken  to  be  the  kinetic  energy  in  bay  N, 
so  that  the  aim  is  to  minimise  the  vibration  transmitted  along  the  structure. 
Three  types  of  loading  are  considered;  (i)  single  frequency  loading  with  Q=50, 


1116 


which  lies  within  the  second  pass  band  of  the  ordered  structure;  (ii)  band- 
limited  loading  with  40<Q<60,  which  covers  the  whole  of  the  second  pass 
band;  (iii)  band-limited  loading  with  23<Q<61,  which  covers  the  whole  of  the 
second  stop  band  and  the  second  pass  band. 

Results  for  the  optimal  design  under  single  frequency  loading  are  shown  in 
Table  1;  in  all  cases  it  was  found  that  the  bay  lengths  were  placed  against 
either  the  upper  bound  (U=1.1L,)  or  the  lower  bound  (L=0.9L,),  and  significant 
reductions  in  the  energy  level  of  bay  N  were  achieved.  In  this  regard  it  should 
be  noted  that  the  dB  reduction  quoted  on  Table  1  is  defined  as  -101og(r;v/^;^r) 
where  is  the  kinetic  energy  in  the  final  bay  of  the  ordered  system.  The 
optimal  designs  shown  in  Table  1  all  tend  to  consist  of  a  bi-periodic  structure 
in  which  the  basic  unit  consists  of  two  bays  in  the  configuration  LU.  The  pass 
bands  and  stop  bands  for  this  configuration  are  shown  in  Figure  3,  and  further, 
for  the  optimal  12  bay  system  is  shown  in  Figure  4  over  the  frequency 
range  0<Q<250.  By  comparing  Figures  3  and  4  it  is  clear  why  the  selected 
design  is  optimal  -  the  new  bi-periodic  system  has  a  stop  band  centred  on  the 
specified  excitation  frequency  Q=50.  It  can  be  seen  from  Figure  4  that  the 
improvement  in  the  response  at  the  specified  frequency  0=50  is  accompanied 
by  a  worsening  of  the  response  at  some  other  frequencies. 

Results  for  the  optimal  design  under  band-limited  excitation  over  the  range 
40<Q<60  are  shown  in  Table  2.  In  some  cases  two  results  are  shown  for  the 
optimised  "Final  Energy":  in  such  cases  the  first  result  has  been  obtained  by 
forcing  each  bay  length  onto  either  the  upper  (U)  or  lower  (L)  bound,  while 
the  second  result  has  been  obtained  by  using  the  NAg  optimisation  routine. 
If  only  one  result  is  shown  then  the  two  methods  yield  the  same  optimal 
design.  The  "bound"  result  is  easily  obtained  by  computing  the  response  under 
each  possible  combination  of  U  and  L  bay  lengths  -  this  requires  2^  response 
calculations,  which  normally  takes  much  less  CPU  time  than  the  NAg 
optimisation  routine.  It  is  clear  from  Table  2  that  the  additional  improvement 
in  the  response  yielded  by  the  full  optimisation  routine  is  minimal  for  this 
case.  The  response  curve  for  the  12-bay  system  is  shown  in  Figure  5,  where 
it  is  clear  that  a  significantly  reduced  response  is  achieved  over  the  specified 
frequency  range;  as  would  be  expected  an  increase  in  the  response  can  occur 
at  other  frequencies.  It  is  interesting  to  note  that  most  of  the  optimal  designs 
shown  in  Table  2  lack  symmetry  -  however,  it  follows  from  the  principle  of 
reciprocity  that  a  design  which  minimises  vibration  transmission  from  left  to 
right  will  also  minimise  transmission  from  right  to  left.  It  should  therefore  be 
possible  to  "reverse"  the  designs  without  changing  the  transmitted  vibration 
levels.  This  hypothesis  is  tested  in  Figure  6  for  a  12  bay  structure  -  the  figure 
shows  the  energy  distribution  for  the  optimal  design  UUULUULLLLLU  and 
for  the  reversed  design  ULLLLLUULUUU.  Although  the  detailed  distribution 
of  energy  varies  between  the  two  designs,  the  energy  levels  achieved  in  bay 


1117 


12  are  identical,  as  expected. 


Results  for  the  optimal  design  under  wide-band  excitation  23<Q<61  are  shown 
in  Table  3,  and  the  response  curve  for  the  12-bay  optimised  system  is  shown 
in  Figure  7.  The  form  of  optimal  design  achieved  is  similar  to  that  obtained 
for  the  narrower  excitation  band  40<Q<60,  although  there  are  detailed 
differences  between  the  two  sets  of  results.  In  each  case  there  is  a  tendency 
for  a  group  of  lower  bound  bays  (L)  to  occur  in  the  mid  region  of  the 
structure,  and  a  group  of  upper  bound  bays  (U)  to  occur  at  either  end.  This 
creates  an  "impedance  mismatch"  between  the  two  sets  of  bays,  which 
promotes  wave  reflection  and  thus  reduces  vibration  transmission  along  the 
structure.  By  comparing  Tables  1-3,  it  is  clear  that  the  achievable  reduction 
in  vibration  transmission  reduces  as  the  bandwidth  of  the  excitation  is 
increased. 

3.3  Design  for  Minimum  ''Maximum”  Strain  Energy 

In  this  case  the  strain  energy  U„  of  each  bay  is  computed  and  the  objective 
function  is  taken  to  be  the  maximum  value  of  U„.  As  a  design  objective,  this 
procedure  can  be  likened  to  minimising  the  maximum  stress  in  the  structure. 
As  in  the  previous  section  the  three  frequency  ranges  Q=50,  40<n<60,  and 
23<Q<61  are  considered,  and  the  present  study  is  limited  to  systems  having 
9,10,  11,  or  12  bays;  the  optimal  designs  achieved  are  shown  in  Table  4. 

Considering  the  single  frequency  results  (Q.=50)  shown  in  Table  4,  it  is  clear 
that  a  large  dB  reduction  is  achieved  only  for  those  systems  which  have  an 
even  number  of  bays;  furthermore,  the  optimal  energy  obtained  has  the  same 
value  (0.0297)  in  all  cases.  This  can  be  explained  by  noting  that  for  an  odd 
number  of  bays  the  frequency  Q=50  lies  near  to  an  anti-resonance  of  the 
ordered  structure,  whereas  a  resonance  is  excited  for  an  even  number  of  bays  - 
this  feature  is  illustrated  in  Figure  8  for  the  12  bay  structure.  The  repeated 
occurrence  of  the  optimal  energy  0.0297  arises  from  the  fact  that  the  initial  bay 
pattern  ULLLUUU  occurs  in  all  four  designs  -  it  has  been  found  that  this 
pattern  causes  a  vibration  reduction  of  over  20dB  from  bay  1  to  bay  8,  so  that 
the  response  in  bay  1  (the  maximum  response)  is  insensitive  to  the  nature  of 
structure  from  bay  8  onwards. 

The  optimal  "bounded"  designs  arising  for  band-limited  excitation  either  tend 
to  be  of  the  "UL"  bi-periodic  type  or  else  nearly  all  the  bays  are  assigned  the 
same  length.  However  it  should  be  noted  that  in  all  cases  the  design  produced 
by  the  NAg  optimisation  routine  offers  an  improvement  over  the  "bounded" 
design,  particularly  for  the  wide-band  case  (23<n<61).  It  is  clear  from  Table 
4  that  the  achieved  reduction  in  strain  energy  reduces  as  the  bandwidth  of  the 
excitation  is  increased. 


1118 


4.  CONCLUSIONS 


The  present  work  has  considered  the  optimal  design  of  a  near-periodic  beam 
system  to  minimise  vibration  transmission  and  also  maximum  stress  levels. 
With  regard  to  vibration  transmission  it  has  been  found  that  very  significant 
reductions  in  transmission  are  achievable  with  relatively  minor  design  changes. 
The  optimum  design  normally  involves  placing  the  design  parameters  (the  bay 
lengths)  on  the  permissible  bounds,  and  this  means  that  a  simple  design  search 
routine  can  be  used  in  preference  to  a  full  optimisation  algorithm.  With  regard 
to  minimum  stress  levels,  it  has  been  found  that  the  optimal  design  for  wide¬ 
band  excitation  is  not  normally  a  "bounded"  design,  and  thus  use  of  a  full 
optimisation  algorithm  is  preferable  for  this  case.  For  both  vibration 
transmission  and  maximum  stress  levels,  the  benefits  obtained  from  an  optimal 
design  decrease  with  increasing  excitation  bandwidth,  but  nonetheless  very 
significant  reductions  can  be  obtained  for  wide-band  excitation. 

REFERENCES 

1.  S.S.  MESTER  and  H.  BENAROYA  1995  Shock  and  Vibration  2,  69- 
95.  Periodic  and  near-periodic  structures. 

2.  D.J.  MEAD  1996  Journal  of  Sound  and  Vibration  190,  495-524.  Wave 
propagation  in  continuous  periodic  structures:  research  contributions 
from  Southampton  1964-1995. 

3.  C.H.  HODGES  1982  Journal  of  Sound  and  Vibration  82,  411-424. 
Confinement  of  vibration  by  structural  irregularity. 

4.  D.  BOUZIT  and  C.  PIERRE  1992  Journal  of  Vibration  and  Acoustics 
114,  521-530.  Vibration  confinement  phenomena  in  disordered,  mono- 
coupled,  multi-span  beams. 

5.  R.S.  LANGLEY  1996  Journal  of  Sound  and  Vibration  189,  421-441. 
The  statistics  of  wave  transmission  through  disordered  periodic 
waveguides. 

6.  R.S.  LANGLEY  1995  Journal  of  Sound  and  Vibration  188,  717-743. 
Wave  transmission  through  one-dimensional  near  periodic  structures: 
optimum  and  random  disorder. 

7.  N.S.  BARDELL,  R.S.  LANGLEY,  J.M.  DUNSDON  and  T.  KLEIN 
1996  Journal  of  Sound  and  Vibration  197,  427-446.  The  effect  of 
period  asymmetry  on  wave  propagation  in  periodic  beams. 

8.  ANON  1986  The  NAg  Fortran  Workshop  Library  Handbook  -  Release 
1.  Oxford:  NAg  Ltd. 


1119 


TABLE  1 

Optimal  design  of  1-D  beam  structure,  to  minimise  energy  transmission,  D.=50. 


Original  Energy;  Non-dimensional  kinetic  energy  in  bay  N  of  the  periodic  structure. 
Final  Energy;  Non-dimensional  kinetic  energy  in  bay  N  of  the  optimised  structure. 


No.  of 

Optimal  Pattern 

Original 

Final 

Reduction 

Bays,  N 

Energy 

Energy 

(dB) 

4 

UULU 

0.276E  1 

0.804E-3 

35.348 

ULULU 

0.609E-1 

0.179E-3 

■S 

UULULULU 

0.674E  0 

0.613E-5 

ULULULULU 

0.564E-1 

0.135E-5 

46.216 

1— 

10 

UULULULULU 

0.424E  0 

0.532E-6  ^ 

11 

ulululululu 

0.535E-1 

0.n7E-6 

56.604 

12 

UULULULULULU 

0.289E  0 

0.461E-7 

67.966 

13 

ULULULULULULU 

0.502E-1 

O.lOlE-7 

66.950 

16 

UULULULULULULULU 

0.154E0 

0.346E-9 
_ _ — 

86.484 

17 

ululululululululu 

0.43  lE-1 

0.761E-10 

87.529  1 

1120 


TABLE  2 

Optimal  design  of  1-D  beam  structure,  to  minimise  energy  transmission,  40<Q<60. 

Original  Energy:  Non-dimensional  kinetic  energy  in  bay  N  of  the  periodic  structure. 

Final  Energy:  Non-dimensional  kinetic  energy  in  bay  N  of  the  optimised  structure. 


No.  of 

Optimal  pattern 

Original  | 

Final 

Reduction 

Bays,  N 

Energy 

Energy 

(dB) 

■■ 

ULLU 

0.670E  0 

0.103E-1 

18.112 

5 

ULLLU 

0.63  IE  0 

0.735E-2 

19.338 

0.711E-2 

19.482 

6 

UULLLU 

0.221E-2 

22.407 

7 

ULLUULU 

0.463E  0 

0.171E-2 

24.335 

8 

UULLLLLU 

0.430E  0 

0.966E-3 

26.487 

0.914E-3 

26.725 

9 

UUULLLLLU 

0.444E  0 

0.341E-3 

31.142 

10 

UUUULLLLLU 

0.449E  0 

0.192E-3 

33.681 

0.189E-3 

33.758 

11 

ULLUUUULLLU 

0.291E0 

0.821E-4 

35.504 

12 

UUULUULLLLLU 

0.201E0 

0.352E-4 

37.558 

13 

1  ULUUUUULLLLLU 

0.199E0 

0.153E-4 

41.148 

1121 


TABLE  3 

Optimal  design  of  1-D  beam  structure,  to  minimise  energy  transmission,  23<Q<6L 

Original  Energy;  Non-dimensional  kinetic  energy  in  bay  N  of  the  periodic  structure. 

Final  Energy:  Non-dimensional  kinetic  energy  in  bay  N  of  the  optimised  structure. 


6 

LLLLUU 

7 

LLLULUU 

8 

LLLLLUUU 

9 

ULLLLLLUU 

0.494E  0 


0.183E0 

0.175E0 


0.648E-2 


0.246E-2 

0.180E-2 


III  » 


UUULLLLLUU 


0.105E0 


0.277E-3 


2.762 

3.916 


18.821 


18.715 

19.878 


21.868 


25.787 


11 

UUULLLLLLUU 

0.105E0 

0.776E-4 

31.313 

12 

UlTULLLLLLLUU 

0.166E0 

0.526E-4 

34.991 

13 

UUULLLLLLULUU 

0.973E-1 

0.282E-4 

35.379 

14 

UUUULLLLLLULUU 

0.581E-1 

0.122E-4 

36.778 

TABLE 4 

Optimal  design  ofl-D  beam  structure,  to  minimise  "maximum”  strain  energy. 


Bay  No.:  Bay  in  which  the  optimal  minimum  “maximum”  non-dimensional  strain  energy  occurs 

Original  Energy:  Initial  “maximum”  non-dimensional  bay  strain  energy  of  the  periodic  structure. 

Final  Energy:  Non-dimensional  strain  energy  in  bay  N’  of  the  optimised  structure. 


No  of 

Optimal 

Original 

Final 

Bay 

Reduction 

Bays,  N 

Pattern 

Energy 

Energy 

No.,  N’ 

(dB) 

Q.=50 

9 

ULLLUUUUL 

0.667E-1 

0.297E-1 

1 

3.514 

0.296E-1 

1 

3.528 

10 

ULLLUUUULU 

0.540E  0 

12 

ULLLULUULULU 

0.404E  0 

0.297E-1 

1 

11.336 

0.296E-1 

1 

11.351 

40<Q<60 

9 

UUUUULULU 

0.486E  0 

0.710E-1 

1 

8.354 

0.449E-i 

1-2 

10.344 

10 

ULULULULLL 

0.606E  0 

0.643E-1 

1 

9.743 

0.45  IE- 1 

1-2 

11.283 

11 

ULULULULLUU 

0.456E  0 

0.682E-1 

I 

8.252 

0.425E-1 

1-2 

10.306 

12 

UUUIJUUUXJUUUL 

0.332E  0 

0.550E-1 

2 

7.808 

0.412E-1 

1-2 

9.062 

23<n<61 

9 

LLLLLLLLL 

0.234E  0 

0.203E  0 

1 

0.617 

0.979E-1 

1 

3.784 

n 

LLLLLLLLLL 

0.200E  0 

0.178E0 

1 

0.506 

■I 

0.95  IE- 1 

1-2 

3.228 

11 

UUUUUUUUULL 

0.198E0 

0.193E0 

1 

O.lll 

0.910E-1 

1-2 

3.376 

12 

UUUUUOUIJULUU 

0.314E0 

0.182E0 

I 

2.369 

0.803E-1 

1-2 

5.922 

1123 


Phase  £  AUenualion 


Figure  1;  A  simply  supported  periodic 


Itlijsj 


kinelic.enorgy  in  bay  12,  T, 


1128 


EFFECTS  OF  GEOMETRIC  ASYMMETRY  ON  VIBRATIONAL 
POWER  TRANSMISSION  IN  FRAMEWORKS 

J  L  Homer 

Department  of  Aeronautical  and  Automotive  Engineering 
and  Transport  Studies,  Loughborough  University 
Loughborough,  Leics  ,  LEll  3TU,  UK 


ABSTRACT 

Many  sources,  such  as  machines,  are  installed  on  supports,  or  frameworks, 
constmcted  from  beam-like  members.  It  is  desirable  to  be  able  to  predict 
which  wave  types  will  be  present  at  particular  points  in  the  support  structure. 
By  using  the  concept  of  vibrational  power  it  is  possible  to  compare  the 
contributions  from  each  wave  type.  Wave  motion  techniques  are  used  to 
determine  the  expressions  for  vibrational  power  for  each  of  the  various  wave 
types  present.  The  results  from  the  analysis  show  the  amount  of  vibrational 
power  carried  by  each  wave  type  and  the  direction  of  propagation. 
Consideration  is  given  to  the  effect  on  the  vibrational  power  transmission  of 
introducing  misalignment  of  junctions  in  previously  symmetric  framework 
structures.  By  splitting  a  four  beam  junction  in  to,  say,  a  pair  of  three  beam 
junctions  separated  by  a  small  distance,  it  is  possible  to  establish  the  effects  of 
separating  the  junctions  on  the  various  transmission  paths.  Unlike  other 
techniques  using  vibrational  power  to  analysis  frameworks,  the  model  keeps 
the  contributions  from  each  of  the  various  wave  types  separate.  This  allows 
decisions  to  be  made  on  the  correct  vibrational  control  techniques  to  be 
applied  to  the  structure. 


INTRODUCTION 

When  attempting  to  control  vibration  levels  transmitted  from  a  machine 
through  the  various  connections  to  the  structure  upon  which  it  is  mounted,  it  is 
desirable  to  be  able  to  identify  and  quantify  the  vibration  paths  in  the  stmcture. 
Often  large  machinery  installations  are  installed  on  frameworks  consisting  of 
beam  like  members.  These  frameworks  are  then  isolated  from  the  main 
structure.  Simple  framework  models  are  also  used  in  the  initial  design  stages 
of  automotive  body  shell  structures  to  determine  dynamic  responses. 

If  the  dominant  transmission  path  in  the  framework  is  identified  it  is 
possible  to  reduce  vibration  levels  by  absorbing  the  mechanical  energy  along 
the  propagation  path  in  some  convenient  manner.  By  utilising  the  concept  of 
vibrational  power  it  is  possible  to  quantitatively  compare  the  wave  type 
contributions  to  each  transmission  patL  In  order  to  predict  vibrational  power 
transmission  in  a  framework,  it  is  necessary  to  identify  the  wave  amplitude 
reflection  and  transmission  coefficients  for  each  joint  in  the  structure.  Lee  and 
Kolsky  [1]  investigated  the  effects  of  longitudinal  wave  impingement  on  a 
junction  of  arbitrary  angle  between  two  rods.  Similarly  Doyle  and  Kamle  [2] 
examined  the  wave  amplitudes  resulting  from  a  flexural  wave  impinging  on 
the  junction  between  two  beams.  By  using  the  reflection  and  transmission 
coefficients  for  different  joints,  it  is  possible  to  predict  the  vibrational  power 
associated  with  flexural  and  longitudinal  waves  in  each  section  of  the 


1129 


framework.  Previous  investigations  [3,4]  have  considered  the  effects  of  bends 
and  junctions  in  infinite  beams.  This  work  was  extended  to  consider  the  finite 
members  which  constitute  frameworks  [5].  Unlike  other  techniques  [6,  7] 
utilising  energy  techniques  to  analyse  frame-works,  the  technique  produces 
power  distributions  for  each  wave  type  present  in  the  structure.  By  comparing 
the  results  for  each  wave  type,  it  is  possible  to  apply  the  correct  methods  of 
vibration  control. 

The  technique  is  used  to  investigate  the  effect  of  geometric  asymmetry 
on  the  vibration  transmission,  due  to  steady  state  sinosodial  excitation,  in  a 
framework  structure  similar  to,  say,  those  used  in  the  automotive  industry 
(figure  1).  By  splitting  a  four  beam  junction  into  a  pair  of  three  beam 
junctions  separated  by  a  known  distance,  it  is  possible  to  establish  the  effect  of 
junction  separation  on  the  dominant  transmission  paths.  The  investigation 
presented  is  limited  to  one  dimensional  bending  waves  and  compressive  waves 
only  propagating  in  the  structure.  To  consider  the  addition  of  other  wave 
types  ie.  torsional  waves  and  bending  waves  in  the  other  plane,  the  analysis 
presented  here  for  the  junctions  should  be  extended  as  indicated  by  Gibbs  and 
Tattersall  [3]. 


TRANSMITTED  POWER  IN  A  UNIFORM  BEAM 

For  flexural  wave  motion,  consider  a  section  of  a  uniform  beam  carrying  a 
propagating  flexural  wave.  Two  loads  act  on  this  beam  element,  the  shear 
force  and  the  bending  moment.  It  is  assumed  that  the  flexural  wave  can  be 
described  by  using  Euler-Bemoulli  beam  theory,  so  that  the  displacement  can 
be  expressed  as 

W(x,t)  =  Af  sin  (cot-kfx), 

the  shear  force  acting  on  a  section  as 

S  =  Eia3W/ax3, 

and  the  bending  moment  on  the  section  as 
B  =  El  02  W/ax2. 

Then  the  instantaneous  rate  of  working  X  at  the  cross-section  is  given 
by  the  sum  of  two  terms  (negative  sign  merely  due  to  sign  convention). 

0t  0x0t  0X^  0X^  3x0t 

The  time  averaged  power 
{P)f  =  (1/T)  I  X  dt  then  is  given  by  (P)f  =  Elkf  coAf 

I  (1) 

For  longitudinal  wave  motion  consider  a  section  of  a  uniform  beam 
with  a  longitudinal  wave  propagating  through  the  beam 

U  (x,  t)  =  Ai  sin  (0)t-kix) 


1130 


The  instantaneous  rate  of  working  X  is  then 
X=  -EA(au/ax)u 


and  the  time  averaged  power  is 


Xdt  =  rEA0)kiA| 


(2) 


If  dissipation  is  present  in  the  structure,  the  modulus  of  elasticity  may 
be  considered  to  be  a  complex  quantity 

=  E(l  +  iTi) 


where  represents  the  loss  factor  of  the  material,  present  due  to 
inherent  material  damping. 

The  displacement  of  a  beam  at  a  distance  x  from  the  source,  due  to 
flexural  wave  motion  may  now  be  considered  to  be,  assuming  that 
material  damping  is  small. 


1  ^ 
-kri- 


W  =  Afe  4 

and  the  resulting  time  averaged  power  is  given  by 
(P}f  =  EIcok^e-'‘'’i2  Af 


(3) 


The  above  reduces  to  equation  (1)  at  the  source. 

Similarly,  the  displacement  of  beam,  due  to  longitudinal  wave  motion 
may  be  considered  to  be 

T  ^ 

-kiTi- 

U  =  Aie  ^  sin(cot-kix) 

and  the  resulting  time  averaged  longitudinal  power  may  be  rewritten  as 


(P),=iEAcok,e-‘''^’‘A? 


(4) 


WAVE  TRANSMISSION  THROUGH  A  MULTI  BRANCH  JUNCTION 

Consider  a  four  branch  junction  as  shown  in  figure  2.  Assuming  only  flexural 
and  longitudinal  waves  propagating  in  the  structure,  the  displacements  of  Arm 
1  will  be,  where  A4  represents  the  impinging  flexural  wave  arriving  from 
infinity. 


Wi(x,t)  =  (Aie^f’""  +  A3e‘^fi''  +  A4e  e‘“^ 


(5) 


1131 


(6) 


U,(x,t)  =  (Aae''‘‘i’‘)e‘“‘ 


Similarly  for  arms  2  to  4  the  displacement  will  be, 

where  ^  cos  0n  and  n  is  the  beam  number 

W„(v„,t)  =  )e‘“‘ 

(V) 

(8) 

Here  A3,  A4,  64^  are  travelling  flexural  wave  amplitudes',  Ai  and 
B2n  are  near  field  wave  amplitudes  and  Aa  and  are  travelling 
longitudinal  wave  amplitudes. 

In  previous  work  [2]  in  this  field  a  theoretical  model  was  used  in  which 
it  was  assumed  that  the  junction  between  the  beams  was  a  rigid  mass.  The 
mass  or  joint  is  modelled  here  as  a  section  of  a  cylinder.  This  represents  the 
physical  shape  of  most  joints  in  practical  systems.  It  has  been  shown  [4]  that 
the  joint  mass  has  an  insignificant  effect  on  the  reflected  and  transmitted 
power  for  the  range  of  values  used  in  this  work. 

The  joint  mass  Mj  =  pjTtL^J^/  4,  and  the  moment  of  inertia  of  the  joint 
isIj=ML2/8. 

By  considering  the  conditions  for  continuity  and  equilibrium  at  the 
beam  junction  the  following  expressions  may  be  written. 

For  each  arm 

For  continuity  of  longitudinal  displacement 

L  3W 

Ui=Un  COsen-WnSinen  +  -^^sinen 

For  continuity  of  flexural  displacement 

W,=u„sinen+W„cos0„-|  |^(l  +  cose„) 

For  continuity  of  slope 

8W;  ^  awn 


1132 


For  the  junction 
Equilibrium  of  forces 


'  '  2  '  '  3x2  J 


-vfp  fax  L32w„ 

1 1  I  aVS  2  3v|;^ 


El  Ai  ^  +  Mj  ^  =  i[E„  A„^cose„  +E„  I„^^sine„ 
3x  J  dt^  il^  3V„  3< 


E  T  a^W,  3  r„,  L3W| 

El  Ii  — t^  +  M;  — T  Wi-  — — — 


- ^  -I-  iVi ;  — y  1  “  “  ^T" 

3x2  2  3x 


n  ;^TI  ^  W 

=  Z  En  Ajj-  ”Sin6n-EnIn  2*^  COS0n 

1  I  5¥n 


WAVE  MOTION  AT  A  FORCED  OR  FREE  END 

As  indicated  in  figure  1 ,  the  framework  has  one  forced  end  and  one  free  end. 
Assuming  the  structure  is  only  excited  by  a  transverse  harmonic  force,  the 
boundary  conditions  are  as  follows: 

at  the  forced  end 


E«  =  Pe^«t 

3x2 


EI^  =  0 

3x2 


EA^  =  0 


Similarly  at  the  free  end  the  above  boundary  conditions  apply  with  the 
exception  that 

EI^  =  0 
3x2 


1133 


POWER  TRANSMISSION  THROUGH  A  FRAMEWORK 


The  structure  shown  in  figure  1  consits  of  one  four-beam  junction,  two  three- 
beam  junctions  and  four  two-beam  junctions.  From  the  equations  detailed  in 
the  above  two  sections,  it  is  possible  to  construct  matrices  of  continuity  and 
equilibrium  equations  for  sub  structures.  These  may  be  combined  to  obtain 
the  overall  matrix  for  the  system.  For  the  framework  shown  in  figure  1,  the 
size  of  the  overall  matrix  is  60x60.  This  matrix  may  be  solved  to  obtain  the 
sixty  unknown  wave  amplitude  coefficients  from  which  time  averaged 
transmitted  power  for  each  beam  may  be  calculated  using  equations  (3)  and 
(4). 

Normalised  nett  vibrational  power  is  then  calculated  at  the  centre  of 
each  beam  constituting  the  structure.  Nett  vibrational  power  may  be 
considered  to  be  the  difference  between  power  flowing  in  the  positive 
direction  and  power  flowing  in  the  negative  direction  for  each  wave  type. 
Normalised  nett  power  is  considered  to  be  nett  power  divided  by  total  input 
power.  The  input  power  to  a  structure  may  be  calculated  from  the  following 
expression  [8] 

Input  Power  =  ^  IFI IVI  cos0 


where  0  is  the  phase  angle  between  the  applied  force  and  the  velocity 
of  the  structure  at  the  forcing  position. 


Figures  3-6  show  the  nett  normalised  power  in  each  arm  of  a 
framework  structure  over  the  frequency  range  0-lkHz  excited  by  IN  force, 
whose  material  and  geometric  properties  are  given  in  Appendix  2.  For  the 


results  shown,  angle  1  is  45®  and  angle  x  is  40®  (or  the  ratio 


=  0.89)  and 


L  =  y  =  0.1m.  Using  these  parameters  the  ratio  of  the  length  of  beam  No.6  to 
beam  No.4  is  0.12.  The  predicted  flexural  power  is  shown  in  figures  3  and  4 
and  from  these  it  can  be  seen  that  the  dominant  transmission  paths  are  arms  1 
and  5,  the  forced  and  free  arms.  The  transmitted  power  in  arm  10  is  next 
dominant  and  comparable  to  arm  5  in  the  region  0-600Hz. 

The  response  for  ail  other  arms  are  small,  typically  less  than  5%  of 
input  power,  with,  as  would  be  expected,  arms  2  and  9  being  approximately 
identical  in  transmission  properties. 

Figures  5  and  6  show  the  nett  normalised  power  for  the  longitudinal 
waves  in  the  structure.  As  the  frequency  range  of  interest  corresponds  to  a 
flexural  Helmoltz  number  of  1  to  5  with  L  being  the  reference  length,  the 
conversion  of  power  from  flexural  to  longitudinal  waves  is  minimal.  From  the 
figures  it  can  be  seen  that  beams  1,  5,  6  and  10  have  identical  transmission 
characteristics,  which  would  be  expected  at  such  large  longitudinal 
wavelengths.  Significant  longitudinal  power  is  only  observed  in  arms  3  and  8 
in  the  frequency  region  200-300Hz.  This  frequency  region  coincides  with  a 
drop  in  the  flexural  power  due  to  the  structure  being  at  resonance  in  that 
region.  It  should  be  noted  that  power  transmitted  through  arms  3  and  8  has 
travelled  through  two  junctions. 


1134 


EFFECT  OF  GEOMETRIC  ASYMMETRY 


By  altering  the  ratio  of  angle  6i  to  angle  0x  it  is  possible  to  alter  the  length  of 
beam  6  and  hence  move  a  pair  of  three  arm  junctions  further  or  closer  apart. 
From  the  discussion  in  the  previous  section,  it  was  seen,  for  the  structure 
under  investigation,  that  the  dominant  flexural  path,  not  surprisingly,  is 
through  the  centre  of  the  structure,  whilst  the  peaks  in  longitudinal  power 

occur  in  beams  3  and  8.  Thus  0x  was  varied  and  the  effect  on  transmission  in 
the  dominant  paths  noted. 

Figures  7-9  show  flexural  power  for  arms  5  and  10  and  longitudinal 
power  for  arm  8  for  four  values  of  0x.  The  values  chosen  were  36°,  38.25°, 
40°  and  42.75°  which  are  equivalent  respectively  to  0x  over  0]  ratios  of  80%, 

85%,  90%  and  95%.  Thus  as  0x  increases,  the  structure  moves  to  being 
symmetrical  in  nature.  From  figure  7,  the  increase  in  junction  separation 
decreases  power  in  the  frequency  region  0-500Hz  and  increases  it  in  the  region 
500-lkHz.  In  beam  10  (figure  10)  the  effect  on  the  flexural  power  is  reversed 
with  increase  in  junction  separation  leading  to  increased  power  below  500Hz 
and  decreased  power  above  500Hz.  It  should  also  be  noted  that  increased 
junction  separation  has  little  effect  on  the  power  below  250Hz.  The  effect  was 
also  noted  on  all  other  beams  which  had  both  ends  connected  to  a  joint.  It 
may  be  concluded  that  at  long  flexural  wavelengths  the  junction  separation  has 
little  effect  with  the  impedance  mis-match  at  the  junctions  being  the  important 
criteria  to  effect  transmission.  It  should  also  be  noted  that  increasing  power  in 
one  arm  ie.  5,  causes  a  decrease  in  arms  ie.  10,  connected  to  it.  An  example  of 
the  effect  of  junction  separation  on  longitudinal  power  is  shown  in  figure  9. 
This  shows  nett  normalised  longitudinal  power  for  arm  8  for  the  same 

variation  in  0x.  Again  minimal  effect  is  seen  at  low  frequencies,  with 
increased  junction  separation  having  different  effects  in  different  frequency 
region.  Increasing  junction  separation  has  little  effect  on  the  region  between 
200  and  300Hz  when  the  longitudinal  power  was  dominant.  This  would  be 
caused  by  the  junction  separation  having  little  effect  on  the  structures  flexural 
natural  frequencies.  Only  by  shifting  those  would  the  peaks  in  longitudinal 
power  by  shifted  in  frequency. 


CONCLUSIONS 

Results  are  presented  for  normalised  nett  time  average  vibrational  power  for  a 
framework  structure.  The  geometric  symmetry  of  the  structure  is  broken  by 
allowing  one  angle  to  decrease  in  value.  The  effects  of  varying  the  angle 
change  by  up  to  20%  of  its  original  value  are  investigated.  Although  the 
results  presented  are  for  one  example  only,  highlighted  are  the  fact  that 
decreases  in  power  in  one  part  of  the  structure  result  in  increases  in  power  in 
another  part.  Also  shown  was  the  effect  of  splitting  a  junction  in  to  a  pair  of 
junctions  is  minimal  at  low  frequencies,  or  long  wavelengths.  From  the  results 
of  the  analysis  it  is  possible  to  establish  frequencies  and  positions  for 
minimum  power  on  the  structure.  Other  configurations  of  framework 
structure  may  be  analysed  by  applying  the  equations  presented. 


1135 


REFERENCES 


1.  J.  P.  LEE.  and  H.  KOLSKY  1972  Journal  of  Applied  Mechanics  39,  809- 
813.  The  generation  of  stress  pulses  at  the  junction  of  two  non-collinear 
rods. 

2.  J.  F.  DOYLE  and  S.  KAMLE  1987  Journal  of  Applied  Mechanics  54, 
136-140.  An  experimental  study  of  the  reflection  and  transmission  of 
flexural  waves  at  an  arbitrary  T-Joint. 

3.  B.  M.  GIBBS  and  J.  D.  TATTERS  ALL  1987  Journal  of  Vibration, 
Acoustics,  Stress  and  Reliability  in  Design,  109,  348-355.  Vibrational 
energy  transmission  and  mode  conversion  at  a  corner  junction  of  square 
section  rods. 

4.  J.  L.  HORNER  and  R.  G.  WHITE  1991  Journal  of  Sound  and  Vibration 
147,  87-103.  Prediction  of  vibrational  power  transmission  through  bends 
and  joints  in  beam-like  structures. 

5.  J.  L.  HORNER  1994  Proceedings  of  the  5th  International  Conference  on 
Recent  Advances  in  Structural  Dynamics,  SOUTHAMPTON  UK,  450- 
459.  Analysis  of  vibrational  power  transmission  in  framework  structures. 

6.  P.  E.  CHO  and  R.  J.  BERNHARD  1993  Proceedings  of  the  4th 
International  Congress  on  Intensity  Techniques,  SENLIS,  France,  347- 
354.  A  simple  method  for  predicting  energy  flow  distributions  in  frame 
structures. 

7.  M.  BESHARA  and  A.  J.  KEANE  1996  Proceedings  of  Inter-Noise  '96, 
LIVERPOOL,  UK  2957-2962.  Energy  flows  in  beam  networks  with 
complient  joints. 

8.  R.J.PINNINGTON  and  R.G.WHITE  1981  Journal  of  Sound  and  Vibration 
75,  179-197.  Power  flow  through  machine  isolators  to  resonant  and  non- 
resonant  beams. 


APPENDIX  1  -  NOTATION 


A 

-  Cross  sectional  area 

Q 

-  Axial  force 

Af 

-  Amplitude  of  flexural  wave 

s 

-  Shear  force 

Ai 

-  Amplitude  of  longitudinal  wave 

T 

-  Time  period 

B 

-  Bending  moment 

t 

-  Time 

E 

-  Young’s  modulus 

U 

-  Displacement  due  to 

E* 

-  Complex  Young’s  modulus 

longitudinal  wave  motion 

F 

-  Excitation  force 

V 

-  Velocity 

I 

-  Moment  of  inertia 

w 

-  Displacement  due  to 
flexural  wave  motion 

Ij 

Jw 

-  Moment  of  inertia  of  joint 

X 

-  Instantaneous  rate  of 
working 

-  Joint  width 

X 

-  Distance 

-  Loss  factor 

kf 

-  Flexural  wave  number 

0n 

-  Angle  of  Arm  n 

ki 

-  Longitudinal  wave  number 

Pj 

-  Joint  density 

L 

-  Joint  length 

M 

-  Moment  force 

Mj 

n 

P 

-  Joint  mass 

<}> 

Phase  angle 

-  Beam  number 

-  Transverse  force 

¥n 

-  Distance  along  Arm  n 

1136 


<P>f  -  Time  averaged  flexural  power  co  -  Frequency  (rad/s) 
<P>1  -  Time  averaged  longitudinal  power 

APPENDIX  2  -  MODEL  PROPERTIES 

33mm 
6mm 
5GN/m2 
1180kg/m3 
0.001 


1 


Beam  Breadth 
Beam  Depth 
Youngs  Modulus 
Density 
Loss  factor 


Figure  1 :  Framework  Structure 


iNett  Normalised  Power 


Nett  Normalised  Power 


Figure  5:  Longitudinal  Power  -  Beams  1-5 


(Beam  1 - ,  Beam  2 . ,  Beam  3  .  .  Beam  4 - ,  Beam  5 


Figure  6:  Longitudinal  Power  -  Beams  6-10 

(Beam  6 - ,  Beam  7 . ,  Beam  8  .  .  Beam  9 - ,  Beam  10 


1139 


THE  INFLUENCE  OF  THE  DISSff  ATION  LAYER  ON  ENERGY 
FLOW  IN  PLATE  CONNECTIONS 


Marek  Iwaniec,  Ryszard  Panuszka 

Technical  University  of  Mining  and  Metallurgy, 
Structural  Acoustics  and  Intelligent  Materials  Group 
30-059  Cracow,  al.  Mickiewicza  30,  Poland 


1.  Introduction 


Dynamic  behaviour  of  mechanical  strucmres  may  be  modelled  on  the  basis  of 
and  with  the  help  of  mathematical  apparams  used  in  Statistical  Energy  Analysis  (SEA) 
[5]  The  method  is  especially  useful  to  calculate  the  statistical  approach  vibroacoustical 
energy  flow  in  middle  and  high  frequency  range.  With  the  help  of  a  few  parameters, 
such'as-  modal  density,  damping  loss  factor,  coupling  loss  factor  and  the  value  of 
input  power,  building  linear  equations  set  it  is  possible  to  describe  the  flow  of 
vibroacoustical  energy  in  a  complicated  stmcture.  There  is  also  a  possibility  of  quick 
estimation  of  the  influence  of  constmction  method  on  the  vibroacoustical  parameters 
of  the  whole  set.  In  the  following  work  an  exemplary  application  of  one  of  the  most 
frequently  used  software  for  calculating  the  flow  of  acoustic  energy  has  been 
presented-  AutoSEA  programme  [1].  The  aim  of  the  work  is  practical  modelling  of 
vibroacoustical  energy  flows  through  screw-connection  of  two  plates  and  comparing 
quantity  results  with  experimental  (outcome)  measurements.  Equivalent  coupling  loss 
factor  has  been  calculated  for  a  group  of  mumally  combined  elements  constimting  a 
construction  fragment.  A  comparison  between  the  measured  results  and  the  value  of 
coupling  loss  factor  in  linear  joint  (e.g.  in  welded  one)  has  also  been  made.  Using  the 
method  of  fmite  elements,  the  influence  of  rubber  separator  thickness  on  the  value  of 
the  first  several  frequencies  of  free  vibrations  has  been  computed  as  well. 

2.  Physical  model  of  plate  connection 

A  connection  of  two  perpendicular  plates  has  been  chosen  for  modelling  the 
flow  of  vibroacoustical  energy  in  mechanical  joints.  Connection  diagram  is  presented 
in  Figure  1.  On  the  length  of  common  edge  the  plates  has  been  joined  with  anglesteel 
by  screws.  A  rubber  separator  (4)  has  been  placed  between  the  excited  plate  and  the 
anglesteel  leg  (3). 


1143 


Modelled  stmcture 


3.  SEA  model 

In  order  to  carry  out  the  vibroacoustical  analysis  of  the  system  using  Statistical 
Energy  Analysis  a  model  of  the  examined  strucmre  has  been  built.  It  has  been 
assumed  that  in  every  element  of  the  construction  only  flexural  waves  propagate. 
Every  plate  and  the  rubber  layer  have  been  modelled  with  just  one  appropriately 
chosen  subsystem.  The  anglesteel,  however,  has  been  modelled  as  a  continuous 
connection  of  two  plates  having  the  dimensions  which  correspond  to  the  anglesteel 
legs  the  plates  themself  being  set  at  the  right  angle. 


Fig.  2.  SEA  model  of  a  system  Fig-  3.  Modal  densities  of  the  subsystems 


1144 


Using  the  SEA  method  we  are  able  to  describe  the  flow  of  the  vibroacousdcal 
energy  in  middle  and  high  frequencies  with  an  algebraic  equation  set.  The  exMined 
system  consists  of  five  simple  subsystems,  of  which  only  one  is  exited  to  vibration 
with  applied  force.  The  flow  of  vibroacoustical  energy  m  the  model  presented  is 
depicted  with  the  following  equation: 


^  1  tot 

-^21 

0 

0 

12 

^  2  tot 

32 

0 

0 

“'•123 

^  3  tot 

-‘n43 

0 

0 

“''134 

h  4  tot 

0 

0 

0 

"'’145 

0 

tot 

w, 

0 

^2  tot 

0) 

0 

0 

• 

^3  tot 

= 

0 

-^54 

^4  tot 

0 

^  5  tot 

^5  tot 

0 

^  ^  4-  is  a  total  coefficient  of  energy  loss  for  every  subsystem, 

rj’.  '"'  -  intemarioss  factor  of  the  subsystem, 

ri..  -  coupling  loss  factor  between  subsystems, 

E-  -  the  mean  vibrational  energy  in  Af  frequency  band  in  i-subsystem, 
W  -  the  input  power  carried  into  i-subsystem  from  outside. 


To  determine  the  elements  of  the  coefficients  matrix  in  eq.  1  it  is  necessary  to 
know  [1,4,6]  coupling  loss  factors  (CLF)  between  structural  subsystems  and  damping 

In  the  SEA  model  in  question  transmission  of  the  acoustic  energy  occurs  in  two 
tvpes  of  connections  between; 

the  plate  and  the  beam  (the  point  joint  of  the  beam  which  is  parallel  to  the  edge 

of  the  plate,  (transfers  flexural  waves),  .  .  „ 

two  plates  (linear  connections  and  point  joints  transferring  flexural  waves). 


The  coupling  loss  factor  between  the  plate  and  the  beam  which  vibrate  in  the 
flexural  way  (in  the  case  of  the  point  joint),  is  defined  with  following  equation  [1]. 


1.75c^ 


(2) 


where: 

c  -  is  the  speed  of  flexural  wave, 

T  -  transmission  factor, 

Q  -  the  number  of  point  connections, 
Gj  -  angular  frequency, 

A  -  the  surface  of  the  plate. 


1145 


The  speed  of  the  flexural  wave  in  the  first  plate  c„  can  be  calculated  in  the 
following  way: 


(3) 


In  the  model  under  examination  formula  (2)  defines  coupling  loss  factors  between  the 
anglesteel  leg  (2)  and  the  beam  (4)  -  ( factors  1,3.  and  tiJ  or  the  beam  (4)  and  the  plate 
(5)  -  rj45  and  7/54. 

The  flow  of  energy  between  two  plates  (which  are  connected  at  the  right  angle 
and  which  vibrate  in  flexural  way)  is  defined  with  the  following  formula  in  the  case 
of  linear  connection: 


where: 

1  -  is  the  length  of  the  connection. 

With  above  formula  it  is  possible  to  describe  the  flow  of  energy  through  correctly 
made  welded  joints  of  plates  or,  for  example,  through  bent  plates.  In  the  system 
presented  the  factor  determines  the  flow  of  energy  between  anglesteel  legs  (2)  and 

(3) 

The  coupling  loss  factor  between  two  plates  with  a  point  joint  is  described  by 
the  following  formula: 


iiL^ 

3  (o.X, 


(5) 


This  type  of  connection  occurs  between  the  plate  representing  the  anglesteel 

leg  and  the  plate  (5).  ^  . 

After  defining  the  value  of  factors  matrix  in  the  first  equation  it  is  possible  to 

specify  the  ratio  of  the  vibroacoustical  energy  gathered  in  plate  (1)  and  (5). 

_  (n  2tof^  3t0t~  ^  24^  32)  4fot^  5tnt~  ^  54^  45^  ~  34^  43^  5tot  (5) 

£5  ^21^  32'^  43^  54 

The  damping  loss  factor  is  important  parameter  of  every  subsystem.  For  steel 
plates  used  in  the  experiment  the  value  of  the  damping  loss  factor  have  been  measured 
experimentally  with  the  decay  method.  The  results  of  the  measurements  have  been 
presented  in  figure  4.  The  frequency  characteristic  of  rubber  damping  has  been  shown 
in  figure  5. 


1146 


DLF  of  steel  [-] 


frequency  [Hz]  frequency  [Hz] 


Fig.  4.  Damping  loss  factor  of  steel  Fig.  5.  Damping  loss  factor  of  rubber 
4.  Experimental  research 

Experimental  investigations  have  been  carried  out  for  connections  made  with 
the  use  of  rubber  separator  (elastic  layer)  of  50°  Shore  hardness.  The  connection  was 
build  up  of  two  identical,  perpendicular  plates  connected  each  to  another  using  the 
anglesteel  and  the  elastic  rubber  layer.  These  are  the  properties  and  material 
parameters  of  individual  elements: 
plates: 

-  material  constructional  steel  (St3); 

-  dimensions:  500  *  500  *  2.2  mm; 

-  Young  modulus:  2,1  10'^  Pa. 
anglesteel: 

-  material:  constructional  steel  (St3); 

-  dimensions:  L  40  40  2.2  mm; 

-  Young  modulus:  2,1  10' ‘  Pa 
elastic  layer: 

-  material:  rubber  50°  Shore 

-  dimensions:  500  40  mm 

-  thickness:  2,  3,  4,  5,  6  mm 

To  avoid  the  loss  of  mechanical  energy  in  the  environment,  during  measure¬ 
ments  the  construction  was  suspended  to  the  supporting  frame  with  three  weightless 
strings  in  such  way  that  only  rigid  body  motions  in  the  plane  perpendicular  to  the  plate 
surface  can  occur.  The  excitation  of  the  wide-band  type  with  constant  power  spectral 
density  was  applied  in  the  symmetry  axis  of  the  plate  (5)  about  20  mm  below  the 
upper  edge  (Figure  1).  During  the  experiment  the  distribution  of  vibrating  velocities 
on  the  plate  surface  was  obtain  by  non  contact  method  using  laser-vibrometry. 


1147 


5.  FEM  model 

Vibration  of  modelled  structure  in  low  frequency  has  been  analyzed  by  toe 
Finite  Element  Method  (FEM).  Several  FEM  models,  was  build  in  order  to  consider 
L  valid  thickness  of  the  mbber  layer.  The  mbber  thickness  has  been  from 

0  mm  to  6  mm.  The  dimensions  and  material  parameters  of  the  plates  and  the 
LSesteel  was  constant.  The  structure  was  fixed  in  four  comers.  Calc^auons  were 
mfde  for  a  division  of  the  strucmre  into  608  elements  of  type  BRI^8.  The  mfluence 
of  the  rubber  thickness  on  the  eigenfrequencies  was  remarkable.  The  results  for  first 
15  eigenfrequencies  are  presented  in  the  table  1 .  In  fig.  was  shown  also  the  changes 
of  the  value  of  natural  frequencies  in  comparison  with  the  natural  frequencies  of 
strucmre  without  applying  the  mbber  layer. 


table  1.  Namral  frequencies  of  connection 


LaDlc  i.  i>aLutai  ai.  —  - 

1  Natural  frequencies  (Hz)  for  various  rubb 

■  ■ .  -  -  • 

er  layer  tlrickness 

0  nun 

2  nun 

3  mni 

4  nun 

5  nun 

127.8 

123.1 

121.5 

119.4 

117.5 

297.5 

235.8 

224.6 

212.6 

204.6 

507.6 

432.5 

391.3 

357.1 

330.0 

1003.2 

566.9 

525.7 

503.7 

491.9 

1282.3 

807.1 

768.9 

733.5 

712.2 

1900.5 

1185.2 

1169.4 

1123.2 

1070.2 

2386.7 

1570.2 

1483.6 

1376.2 

1304.6 

3272.3 

1851.6 

1634.2 

1531.6 

1490.8 

3603.4 

2303.1 

2263.4 

2167.8 

2147.3 

4263.3 

2611.2 

2539.5 

2496.9 

2463.7 

4759.5 

3192.7 

3515 

3110.1 

2913.2 

5438.4 

3926.7 

3874.6 

3626.4 

2937.0 

7627.4 

4110.1 

4105.5 

3639.9 

3026.6 

7691.6 

5070.7 

5137.0 

3670.5 

3111.2 

11543.2 

5547.1 

5503.9 

3721.9 

3181.2 

The  decreasing  of  the  absolute  values  of  the  natural  frequences  is  observed  according 
to  increasing  of  the  thickness  of  the  rubber  layer.  Beginning  from  die  third  of 

°ib  adons  of  the  system  the  decrease  of  the  natural  frequencies  is  almost  constant  for  the 
"layer  2  mm  or  3  mm  and  is  continuously  decreasing  for  rubber  layer  4-6  mm  (See 

fig.  6). 


1148 


Fig.  6.  Changes  of  natural  frequencies 

6.  Comparison  with  experimental  results 

The  equivalent  coupling  loss  factor  defining  the  energy  flow  between  the  plates  has 
been  determined  experimentally  [31  for  a  model  consisting  of  two  subsystems;  two  plates. 
The  coupling  loss  factor  in  such  two  element  model  may  be  specified  by  the  following 

equation: 


1  zast. 


^2  ^1101 

N,  E2.0C 


-  1 


(7) 


The  quotient  of  plate  energies  E,/E,  in  a  two-element  model  is  relevant  to  the  quotient  of 
energy  in  the  first  and  fifth  subsystem  (E./E,)  in  the  five  element  model  presented  m  figure 
0  the  value  of  these  quotient  is  defined  with  the  equation  (6) 

In  the  picture  we  have  presented  the  values  of  the  equivalent  coupling  loss  factor  in 
the  connection.  Individual  points  in  the  diagram  show  the  results  obtained  experimentally. 
The  values  received  in  computer  simulation  have  been  presented  as  a  continuous  diagram. 
The  upper  curve  shows  the  values  of  the  coupling  loss  factor  in  the  joint  before  the 
application  of  the  rubber  layer. 

In  the  frequency  range  above  ca.  125  Hz  we  have  received  a  very  good  comparison 
of  experimental  results  and  computer  simulation  results  performed  with  the  AutoSEA 
software  At  the  frequency  of  about  200Hz  there  occurs  a  local  minimum  of  the  equivalent 
coupling  loss  factor  between  the  plates.  The  value  of  the  minimum  is  essentially  influenced 
by  the  value  of  the  rubber  damping  loss  factor.  The  frequency  (with  the  minimum  CLF)  is 
strongly  influenced  by  the  peak  frequency  of  the  rubber  damptng  curve. 


1149 


o.ooo2ii - 


symulation 


experiment 


welded  plates 


16'  ‘  '40'  '  ’lOO  250  630  1600  4000 

frequency  [Hz] 

Fig.7  Equivalent  CLF  factor  in  the  joint 

7.  Conclusion 

A  way  of  modelling  the  vibroacoustical  energy  flow  with  the  help  of  SEA  method  has 
been  presented  in  the  work.  We  have  examined  the  screw  connection  of  two  plates,  where 
a  rubber  elastic  layer  has  been  applied,  A  comparison  has  also  been  made  between  the  results 
of  computer  simulation  of  the  mechanical  energy  flow  with  SEA  method  and  the  experimental 
results,  and  thus  we  have  noticed  the  good  correlation,  especially  as  far  as  middle  frequencies 

The  joint  modification  through  introduction  of  the  rubber  separator  has  a  remarkable 

impact  on  the  acoustic  energy  flow:  ,  ^ 

-  The  application  of  the  elastic  layer  in  the  Joint  in  question  lowers  the  value  ot 

eauivalent  coupling  loss  factor  in  the  whole  frequency  range. 

-  the  value  of  rubber  damping  factor  has  most  significant  influence  on  the  acoustic 

eneroy  flow  through  connection  in  middle  frequencies  range: 

“  The  minimum  value  of  the  equivalent  coupling  loss  factor  in  a  joint  is  essentially 
influenced  by  the  rubber  separator  damping  loss  factor. 

*  The  frequency  of  minimum  CLF  occurrence  is  strongly  influenced  by  the  peak 
frequency  of  the  rubber  damping  curve. 

The  increasing  thickness  of  the  rubber  layer  produces  on  decreasing  natural  frequencies  of 
the  structure. 


8.  Bibliography: 

III  AutoSEA  -  User  Guide  Vibro-Acoustic  Sciences  Limited  1992. 

121  Fahy  F.J.  Sound  and  Structural  Vibration;  Radiation,  Transmission  and  Response  Academic  Pres 
nTkirtuin  J  Smals  N.  Panuszka  R.  "Method  of  estimating  the  coupling  loss  factor  for  a  set  of 

nlates"  Mechanic,  Technical  University  of  Mining  and  Metallurgy,  Cracow,  10,1991. 

[^41  Lalor,  N.:  The  evaluation  of  SEA  Coupling  Loss  Factors.  Proc.  V  School  Energy  Methods  in 

Vibroacoustics"  -  Supplement,  Krakow-Zakopane  1996.  ,  .  „  „ 

[51  Lyon  R.,  DeJong  R.;  Theory  and  Application  of  Statistical  Energy  Analysis.  Butterworth- 

Heinemann,  Boston,  1995. 


1150 


Variation  Analysis  on  Coupling  Loss  Factor 
due  to  the  Third  Coupled  Subsystem  in 
Statistical  Energy  Analysis 

Hongbing  Du  Fook  Fah  Yap 
School  of  Mechanical  &  Production  Engineering 
Nanyang  Technological  University 
Singapore  639798 


Abstract 

Statistical  Energy  Analysis  (SEA)  is  potentially  a  powerful  method  for 
analyzing  vibration  problems  of  complex  systems,  especially  at  high  frequen¬ 
cies.  An  impoitant  parameter  in  SEA  modeling  is  the  coupling  loss  factor 
which  is  usually  obtained  analytically  based  on  a  system  with  only  two  cou¬ 
pled  elements.  Whether  the  coupling  loss  factor  obtained  in  the  classical  way 
is  applicable  to  a  practical  problem,  which  normally  comprises  of  more  than 
two  elements,  is  of  importance  to  the  success  of  SEA.  In  this  paper,  the  varia¬ 
tion  of  coupling  loss  factor  between  two  subsystems  due  to  the  presence  of  a 
third  coupled  subsystem  is  investigated.  It  is  shown  that  the  degree  to  which 
the  coupling  loss  factor  is  affected  depends  on  how  strong  the  third  subsystem 
is  coupled.  It  also  depends  on  the  distribution  of  the  modes  in  the  coupled  sub¬ 
systems.  This  kind  of  effect  will  diminish  when  the  damping  is  high,  subsys¬ 
tems  are  reverberant,  or  ensemble-average  is  considered,  but  not  for  individual 
cases. 


1  Introduction 

SEA  is  potentially  a  powerful  method  for  analyzing  vibration  and  acoustic  problems 
of  complex  systems,  especially  at  high  frequencies,  because  of  the  simplicity  of 
its  equations  compared  to  other  deterministic  analysis  techniques.  SEA  models  a 
system  in  terms  of  interconnected  subsystems.  The  coupling  parameter  between 
any  two  subsystems  is  characterized  by  a  coupling  loss  factor.  If  the  coupling  loss 
factors  and  internal  (damping)  loss  factors  of  all  subsystems  are  known,  the  power 
balance  equation  (e.g.,  see  [1])  for  each  subsystem  can  be  established.  From  this  set 
of  equations,  SEA  predicts  the  system  response  (due  to  certain  types  of  excitation) 
in  terms  of  the  average  energy  of  every  subsystem.  The  energy  can  in  turn  be  related 
to  other  response  quantities  such  as  mean  velocity  or  strain. 


1151 


Historically,  the  SEA  power  balance  equations  were  initially  derived  from  an 
analysis  of  two  coupled  oscillators  [2,3].  It  has  been  shown  that  the  energy  flow 
between  them  is  directly  proportional  to  the  difference  in  their  uncoupled  modal 
energies.  The  theory  has  then  been  extended  to  systems  with  multi-coupled  sub¬ 
systems  (e.g.,  [4]).  Strictly  this  extension  is  only  applicable  if  certain  assumptions 
are  justified  [4, 5].  Also  the  new  concept  of  indirect  coupling  loss  factor,  which 
is  used  to  represent  the  energy  flow  proportionality  between  the  indirectly  coupled 
subsystems,  is  also  introduced. 

In  practice,  the  indirect  coupling  loss  factors  are  normally  ignored  in  SEA  ap¬ 
plications  because  they  are  very  difficult  to  determine  analytically.  Only  coupling 
loss  factors  between  directly  coupled  substructures  are  considered.  Some  analy¬ 
ses  [6-8]  have  shown  that  this  approximation  may  lead  to  significant  errors  in  the 
predicted  results  if  certain  conditions  are  not  met  in  the  system.  These  conditions 
include  not  only  the  well  known  requirement  of  weak  coupling  between  subsystems 
(e.g.,  see  [6]),  but  also  others,  such  as  given  by  Langley  that  the  response  in  each 
element  must  be  reverberant  [7];  and  by  Kean  that  there  should  be  no  dominant 
modes  (peaks)  inside  the  frequency-averaging  band  [8].  However,  it  is  usually  dif¬ 
ficult  to  know  whether  these  conditions  are  satisfied  for  a  particular  system.  In  fact 
the  above  mentioned  conditions  do  not  always  hold  for  practical  engineering  cases. 
On  this  point  of  view,  the  importance  of  a  coupling  loss  factor  for  describing  the 
coupling  between  indirectly  coupled  subsystems  are  to  be  further  examined. 

A  related  question  is  whether  the  coupling  loss  factors  obtained  from  the  system 
with  only  two  subsystems  can  still  be  applied  when  other  subsystems  are  present. 
Generally,  the  coupling  loss  factor  is  sensitive  to  the  amount  of  overlap  between 
the  modes  of  the  two  coupled  subsystems.  When  additional  subsystems  are  cou¬ 
pled  to  the  original  two-subsystem  model,  the  mode  distributions  of  the  originally 
coupled  two  subsystems  will  be  affected.  The  change  of  mode  distributions  will  fur¬ 
ther  affect  the  modal  overlap  between  the  coupled  two  subsystems  and  finally  the 
coupling  loss  factor  between  them.  However,  general  estimation  methods  for  cou¬ 
pling  loss  factor  assume  that  the  coupling  parameters  between  two  subsystems  are 
not  affected  much  by  the  presence  of  the  other  subsystems.  Therefore  the  conven¬ 
tional  approaches  of  deriving  coupling  loss  factor  are  mostly  based  on  consideration 
of  a  two-subsystem  model  only.  One  method  is  the  wave  approach,  by  which  the 
coupling  loss  factor  used  in  the  SEA  applications  are  derived  analytically  from  aver¬ 
aged  transmission  factors  of  waves  that  are  transmitted  through  a  Junction  between 
semi-infinite  subsystems.  This  method  only  takes  into  account  local  properties  at 
the  joints  and  sometimes  may  be  inaccurate.  Recent  research  [1,7,9-11]  based 
on  the  model  with  two-coupled  subsystems  has  shown  that  the  coupling  parameter 
does  depend  on  other  system  properties,  such  as  damping  loss  factor,  etc.  It  can  be 
argued  that,  if  there  is  a  third  coupled  subsystem,  the  coupling  parameters  between 
the  first  two  subsystems  will  also  depend  on  the  energy  flow  to  the  third  subsys¬ 
tem.  Therefore,  from  a  practical  point  of  view,  the  coupling  loss  factor  estimated 
for  two-coupled  subsystems,  ignoring  the  indirectly  coupled  subsystems,  can  only 
be  of  approximate  value. 


1152 


In  this  paper,  the  variation  of  coupling  loss  factor  between  two  subsystems  due 
to  the  presence  of  a  third  coupled  subsystem  is  studied.  In  the  following  sections, 
the  coupling  loss  factor  is  firstly  expressed  in  terms  of  global  mobility  functions. 
The  exact  solution  of  mobility  functions  is  only  for  simple  structures.  However,  for 
general  structures,  it  can  be  obtained  by  Finite  Element  Analysis  (FEA)  [10, 1 1]. 
The  coupling  loss  factors  obtained  respectively  in  the  cases  with  and  without  the 
third  subsystem  in  the  model  are  compared  for  two  particular  system  configura¬ 
tions,  respectively.  The  system  used  in  this  investigation  is  one-dimensional  simply 
supported  beanis  coupled  in  series  by  rotational  springs.  By  varying  the  spring  stiff¬ 
ness,  the  strength  of  the  coupling  between  the  second  and  the  third  subsystems  can 
be  changed.  It  is  shown  that  the  effect  of  the  third  coupled  subsystem  on  the  cou¬ 
pling  loss  factor  between  the  first  two  coupled  subsystems  depends  on  how  strong 
the  third  subsystem  is  coupled.  For  each  individual  case,  it  is  also  shown  that  this 
kind  of  effect  may  be  positive  or  negative,  depending  on  the  distribution  of  modes 
in  the  coupled  subsystems. 


2  Coupling  Loss  Factor  by  Global  Modal  Approach 


In  this  section,  a  modal  method  is  used  to  derive  coupling  loss  factor  in  a  sys¬ 
tem  with  any  number  of  coupled  subsystems.  The  result  is  then  simplified  for  two 
cases:  (1)  a  three-subsystem  model;  (2)  a  two-subsystem  model  which  is  simply 
substructured  from  the  previous  three-subsystem  model  by  disconnected  the  third 
subsystem. 

For  a  linear  system  which  consists  of  N  coupled  subsystems,  if  “rain-on-the- 
roof”  excitation  [10]  is  assumed  to  be  applied  to  each  subsystem  in  turn,  the  corre¬ 
sponding  response  energy  can  be  expressed  as 

I  I  drdsdu,  (1) 

sith.'ii/sl.Kmi  Mii.bsy.'it.emj 

where  /?/;  is  the  total  time-averaged  response  energy  of  the  subsystem  i.  due  to 
the  excitation  on  the  subsystem  j,  is  the  transfer  mobility  function  be¬ 

tween  the  response  points  r  and  the  excitation  point  H  is  the  averaging  range 
of  frequency,  u.',  in  and  S  are  the  mass  density  and  the  power  spectral  density  of 
excitation.  The  input  power  due  to  the  excitation  is  given  by 

I  RelH(s,.i,L0)]d.sdu  (2) 

Q  sahsyale'inj 


where  is  the  real  part  of  the  point  mobility  at  the  position 

simplicity,  two  terms,  a/,  and  are  defined  as 


a,,  = 


i>,  =  ^  = 


Al 

m;S  j 

Ik 


\  H [r,  s,uj)\~  dr  ds  cl.in 


n  sii.bsy.'iUjvii  /iubfiyslenij 

lle[H{.^,s,ij)]  dsd.u: 

n  subsyslr;mj 


For 


(3) 

(4) 


1153 


The  mobility  function,  H{r,.s,uj),  is  to  be  expressed  in  terms  of  the  global  modes 
of  the  system,  which  can  be  obtained  by  Finite  Element  Analysis  (FEA).  By  the 
principle  of  reciprocity  of  the  mobility  function,  the  relation  of  aij  =  a  ji  always 
holds  regardless  of  the  strength  of  coupling  and  the  magnitude  of  input  power  if  the 
excitation  is  “rain-on-the-roof Theoretically  applying  the  Power  Injection  Method 
[12]  we  can  obtain  the  SEA  equation  as 

n  =  [77]E  (5) 

where  H  =  {Hi,  ila,  -  •  •  ,  and  E  =  {E[,  E-z,  -  -  -  ,  E^r}'^ .  The  SEA  loss 
factor  matrix  [77]  is 


iVl  + 


i.^2 


(7?yV  + 


■niiau 

niiCLiisr 

bi 

b-z 

bN 

■nua-ii 

■nizCL-zN 

bi 

bo 

bjV 

77?.;Va,Yi 

•m.A/a/VY 

^1 

Ih 

b^! 

=  —  B  A“'M- 

UJr 


where,  //;  is  the  internal  loss  factor  for  subsystem  i.,  v/,,  is  the  coupling  loss  factor 
from  subsystem  -I.  to  subsystem  j,  ujc  is  the  central  frequency  of  the  averaging  band 

n. 


A  =  [a,,] 


From  equation  (6),  the  reciprocity  principle  of  the  coupling  loss  factors  can  be  easily 
seen,  due  to  aij  =  aji. 


1 

''■bi 

,  M  = 

L  ’-J 

Vij  n?. ;  b  j  ni  jbj/ (cu^/i  /  2) 

i]ji  niibi  ■niibi/iujc'^l'l) 


Where,  (77?.;6,:)/(u,v7r/2)  is  the  averaged  real  part  of  point  mobility  [4]  and  can  be 
regarded  as  the  generalized  modal  density  of  the  subsystem  i.  Assuming  weak 
coupling  and  light  damping,  it  approximately  equals  to  the  classical  definition  of 
modal  density  [13].  Therefore,  the  relation  given  by  equation  (7)  also  reduces  to  the 
classical  reciprocity  principle. 


1154 


2.1  Substructured  two-subsystem  model 


Figure  1 :  A  general  SEA  model  with  three  coupled  subsystems  in  series 


Consider  a  whole  system  with  three  substructures  coupled  in  series  as  shown  in 
Figure  1 .  If  subsystem  3  is  removed,  the  coupling  loss  factors  between  subsystems 
1  and  2  are  given  in  the  equation 

biCL22  bi(Li-2 

"h '^712  — ^?2L  _  ^  n?. i(fi  1  [fi22  —  '^7.2(^11^22  —  <'^i2^^'2l) 

—  7/12  772+7/21  u-V  _ ^20.21 _  _ ^2(1 1 1 _ 

777 i{an<^f22  “  <^^12f''2l)  7772(^110-22  —  O12O21) 

r  bj  ^to-12  1  (8) 

^  J_  777  ifli|  7772011022 

62O21  ^2 

7771O11O22  777.2  0  22 

The  approximation  in  the  above  equation  is  due  to  011022  »  012O21  when  the  cou¬ 
pling  is  weak.  Manipulating  equation  (8)  with  or  without  using  the  approximation 
both  can  work  out  the  coupling  loss  factors  7/12  and  7/21  as 


(9) 


r  61  6-, 

( 777  1  I  )  ( 77  72  62  )  - 7/2 - 77 

07^-777-10 1  1  07^777.2022 

!  ,  ^2  .  f.  *^1 

77  7  20'7 -  777  [Oi 

"07c7772022  07^-777-1  011 


(10) 


The  equations  are  true  regardless  of  the  strength  of  the  coupling.  It  can  be  seen  that 
7/12  and  7/21  depend  on  the  values  of  the  three  terms  777.,+/,  7//  and  bij{u:^ni-,au).  The 
first  two  are  the  generalized  modal  density  and  the  internal  loss  factor,  or  in  combi¬ 
nation  equivalent  to  modal  overlap  factor.  The  third  one,  by  noting  the  definitions 
of  6;  and  an,  is  the  ratio  of  input  power  to  response  energy  for  the  directly  excited 
subsystem,  i.e.,  the  total  loss  factor  of  subsystem  i.  From  equation  (8),  this  term 
can  be  approximately  expressed  as 


Total  loss  factor 
of  subsystem  i 


b; 

miLOcCLii 


m  +  5I'+-.7 


(11) 


1155 


In  the  classical  wave  approach,  where  semi-infinite  subsystems  are  assumed,  the 
total  internal  loss  factors  becomes 

■n  total,  I  =  m  +  Vn  and  l]iot.al,2  =  112  +  vTi  (^2) 

where  is  the  classical  coupling  loss  factor.  Substituting  equation  (12)  into  equa¬ 
tions  (9)  and  (10),  i]ij  reduces  to  the  classical  iiff,  which  only  depends  on  the  local 
properties  at  the  joints  rather  than  other  properties  of  the  system,  such  as  damping. 
If  the  subsystem  modal  parameters  are  used  to  evaluate  the  term,  bil{ujcmiaii),  then 
the  total  loss  factor  is  Just  the  internal  loss  factor  of  the  subsystem  and  the  cou¬ 
pling  loss  factor  is  equal  to  zero.  This  is  reasonable  because  using  the  uncoupled 
modal  parameters  instead  of  the  coupled  modal  parameters  is  actually  equivalent  to 
removing  the  coupling  between  two  subsystems. 

However,  for  finite  system  where  the  assumption  of  semi-infinity  is  not  justified, 
there  will  be  no  immediate  simplication  for  equations  (9)  and  (10).  Numerically, 
FEA  can  be  employed  to  obtain  the  global  modes  and  then  the  coupling  loss  factor 
can  be  calculated  [10, 1 1]. 


2.2  Full  three-subsystem  model 

Instead  of  substructuring,  consider  the  three-subsystem  model  as  a  whole  system, 
shown  in  figure  1.  Now  the  order  of  equation  (6)  is  reduced  to  3.  With  the  global 
modal  parameters  obtained  from  FEA,  the  coupling  loss  factors  can  be  directly 
evaluated.  However,  when  the  coupling  between  subsystems  is  weak,  the  order- 
reduced  equation  (6)  is  still  able  to  be  simplified.  Matrix  A  may  be  alternatively 
expressed  in  the  form  of 


■  a,  L 

0 

0  ‘ 

■  0 

«L2 

0 

■  0 

0 

«13  ' 

A  - 

0 

(t-ll 

0 

-f 

a\2 

0 

d.23 

+ 

0 

0 

0 

0 

0 

«33  . 

0 

((•23 

0  . 

.  (l.[3 

0 

0 

where  the  terms  on  the  right  side  are  sequentially  defined  as  Ai,  A2  and  A3 .  Under 
the  assumption  of  weak  coupling,  the  non-zero  entries  in  Ai,  A2  and  A3  will  be  of 
the  order  O(t^),  (9(e^)  and  respectively  [14],  The  inverse  of  matrix  A  may 

be  approximately  written  as 

=  Ai”^  —  Ai“^ A2 Ai”^  —  Ai"^ A3 Ai  ^  4- Ai  ^A2Ai  ^A2Ai 


+  ••• 


(14) 


As  an  approximation,  substituting  only  the  first  three  terms  in  equation  (14)  into 
equation  (6)  gives 


[tj]  ^  l/uv- 


/>■ 

bo  f  t’ 12 
rn  I  f(  1 1  a  2  2 
b3(l-\3 

ni[a\  KM. 3 


■ni2a  1 1  (1-22 

b2 

1112(1.22 

63^23 

1112(1.22^(33 


bi(t.\3 
in3(l\\(l.33 
bid.  13 
in.3Ct.2'i((33 

63  ~ 

1((3((33 


(15) 


1156 


Generally,  under  the  condition  of  weak  coupling,  the  indirect  coupling  loss  factors, 
■ihs  and  7731,  are  much  smaller  than  the  direct  coupling  loss  factors  and  the  internal 
loss  factors  [5,7, 15].  The  diagonal  elements  in  [rj]  can  therefore  be  approximated  to 
the  sum  of  internal  loss  factor  and  direct  loss  factor.  It  can  be  shown  from  equation 
(15)  that  equation  (11)  remains  valid  for  three  coupled  subsystems.  But  in  the  three- 
subsystem  case,  the  term,  b,|{uJcm.^au),  is  to  be  evaluated  by  using  the  global  modes 
of  the  three-subsystem  model. 


3  Numerical  Examples  and  Variation  Analysis 

In  this  section,  two  examples  with  different  configurations  are  used  to  show  the  vari¬ 
ation  of  coupling  loss  factor  due  to  the  presence  of  a  third  coupled  subsystem.  The 
coupling  loss  factor  of  the  two-subsystem  model  is  evaluated  by  using  equations  (9) 
and  (10).  For  the  three-subsystem  model,  equations  (6)  and  (15)  are  used.  It  can  be 
shown  that  both  equations  (6)  and  (15)  give  the  same  results  as  the  couplings  are 
weak. 


3.1  Structural  details  and  SEA  model 


Ki 

beam  I  (TJT]  - 

(a)  ^  - Zi. 

beam  1  rnn  beam  2  (TTH  ^ 

(b)  ^  ^ - A  A - 


Figure  2;  A  structural  model  comprising  of  three  beams 

To  begin  example  calculations,  consider  initially  a  two-subsystem  model  (figure  2(a)) 
which  is  two  thin  beams  coupled  through  a  rotational  spring.  The  group  of  flexural 
vibration  modes  of  each  beam  are  taken  as  a  SEA  subsystem.  The  spring  provides 
weak  coupling  between  them  where  only  rotational  moment  is  transmitted.  When 
beam  3  is  connected  at  the  free  end  of  beam  2  to  the  original  two-beam  model, 
a  three-subsystem  model  is  formed  (figure  2(b)).  The  specifications  for  the  three 
beams  are  given  in  table  1.  The  spring  stiffness,  A'2,  is  adjustable  in  order  to  look 
into  the  significance  of  the  effect  of  the  third  subsystem.  There  are  two  cases  where 
the  length  of  beam  2  is:  (i)  L2  -  1.0?72;  (ii)  L2  =  l-lm.  The  spring  constants  at 
the  joints  are  chosen  to  be  weak  enough  to  ensure  that:(a)  the  coupling  loss  factor 
is  much  smaller  than  the  internal  loss  factor;  (b)  the  indirect  coupling  loss  factor  is 
much  smaller  than  the  direct  coupling  loss  factor. 

In  the  global  modal  approach  (see  section  2),  the  modes  of  two-subsystem  model 
and  three-subsystem  model  are  obtained  from  FEA.  In  numerical  simulation,  the 


1157 


Table  1 :  The  specifications  of  the  three  beams 


BEAM 

1 

2 

3 

length  (m) 

2.0 

1.0  &  1.1 

0.7 

width  (mm) 

4 

Thickness  (mm) 

2 

Density  (Kg/m^) 

7890 

Young’s  Modulus  (N/m'^) 

196E+9 

Poisson  Ratio 

0.29 

spring  constant,  A*i  (Nm/rad) 

1.0 

central  frequency  is  200Hz  and  the  averaging  band  is  selected  as  100  ~  300Hz.  In 
order  to  take  into  account  the  contribution  from  the  modes  out  of  the  band,  all  the 
modes  up  to  500Hz  are  extracted  for  evaluating  the  mobility  functions  in  averaging. 
The  modal  loss  factor  is  assumed  to  be  the  same  for  each  modes  used  in  averaging. 
This  means  that  the  internal  loss  factor  is  the  same  for  each  subsystem  and  is  equal 
to  the  modal  loss  factor  [11].  The  results  given  are  plotted  against  the  modal  loss 
factor  in  order  to  show  the  damping  effect  at  the  same  time. 


3.2  Results  and  discussion 

Figure  3  shows  the  identified  coupling  loss  factor  7/12  for  the  case  (i)  {Lo  =  1.0777) 
with  different  stiffness  of  /v'2.  The  case  of  /v'2  =  0  means  that  the  third  subsystem 


Figure  3:  7/12  is  negatively  affected  in  three-subsystem  model 
is  not  present.  It  can  be  seen  that  the  coupling  loss  factor  7^12  is  decreased  in  the  low 


1158 


range  of  damping  while  the  strength  of  the  coupling  between  subsystem  2  and  3  is 
increased.  The  stronger  the  coupling,  the  more  ijn  is  decreased. 

On  the  other  hand,  for  the  case  (ii)  where  L2  =  l.lm,  the  different  results  are 
shown  in  figure  4  where  the  presence  of  the  third  subsystem  would  mainly  increase 
i]i2  in  the  low  range  of  damping.  The  increasing  magnitude  is  also  dependent  on  the 
strength  of  coupling  between  subsystem  2  and  3.  The  explanation  for  the  different 
variation  trends  of  ?]i2  due  to  the  third  coupled  subsystem  between  figure  3  and  4 
will  be  given  later. 


Figure  4;  771  ■;  is  positively  affected  in  three-subsystem  model 

From  figure  3  and  4,  the  effect  of  damping  on  the  coupling  loss  factor  can  also  be 
observed.  In  the  low  damping  region,  increasing  damping  would  increase  coupling 
loss  factor.  After  a  certain  turnover  point,  increasing  damping  would  make  the 
coupling  loss  factor  decrease  and  finally  7712  becomes  convergent  to  a  value.  This 
agrees  with  the  conclusions  drawn  in  [10,  11].  It  is  shown  that,  even  though  the 
length  of  beam  2  has  a  slight  difference  in  figure  3  and  4,  the  converged  values  are 
still  very  close.  Thus,  the  converged  value  seems  not  to  depend  on  the  variation  of 
coupling  strength  at  A'2  and  the  structural  details,  although,  with  the  third  subsystem 
existing  in  the  system,  the  convergent  speed  is  faster.  Therefore,  it  is  reasonable  to 
believe  that  the  converged  coupling  loss  factor  at  sufficiently  high  damping  only 
depends  on  the  property  of  the  joint  rather  than  other  system  properties.  This  joint- 
dependent  property  of  coupling  loss  factor  in  the  high  range  of  damping  accords 
with  the  assumption  in  the  wave  approach.  Here,  it  is  convenient  to  define  the 
convergent  region  in  the  figure  3  and  4  as  the  “joint-dependent  zone”. 

However,  before  the  “joint-dependent  zone”,  coupling  loss  factor  seems  very 
sensitive  to  the  variation  of  damping  loss  factor  as  well  as  the  strength  of  coupling 
between  subsystem  2  and  3.  It  is  because  in  the  low  damping  region  the  system 


1159 


modal  properties  have  been  playing  a  major  role  in  determining  coupling  loss  fac¬ 
tor  [10, 11].  In  general,  the  coupling  loss  factor  represents  the  ability  of  energy 
transmitted  between  subsystems.  It  depends  not  only  on  the  physical  strength  of 
the  coupling  (e.g.,  spring  stiffness  in  the  examples),  but  also  on  the  amount  of  over¬ 
lap  between  the  modes  of  two  connected  subsystems.  The  higher  modal  overlap 
between  the  modes  of  two  connected  subsystem,  the  more  energy  is  transmitted 
between  the  subsystems.  As  a  result,  the  coupling  loss  factor  will  be  higher  even 
though  the  physical  strength  at  the  joint  is  unchanged.  If  the  modes  in  one  subsys¬ 
tem  are  distributed  exactly  the  same  as  those  in  one  another(for  instance,  two  exactly 
same  structures  are  coupled  together),  the  coupling  loss  factor  would  be  varied  to 
the  maximum,  and  vice  versa.  Therefore,  this  region  could  be  likely  defined  as 
“modal-sensitive  zone”. 

In  the  “modal-sensitive-zone”,  the  dependence  of  coupling  loss  factor  on  the 
amount  of  overlap  between  the  modes  of  two  connected  subsystems  has  been  clearly 
shown  in  figures  3  and  4.  For  the  case  (i)  illustrated  in  figure  3,  the  length  of  beam  2 
is  half  of  beam  1.  Due  to  the  characteristic  of  mode  distribution  in  beam  structure, 
the  amount  of  overlap  between  the  modes  of  subsystem  1  and  2  is  more  than  that  in 
the  case  (ii)  shown  in  figure  4,  where  beam  1  is  2  meters  and  beam  2  is  1.1  meters. 
Therefore,  case  (i)  has  higher  coupling  loss  factor  than  case  (ii)  in  “modal-sensitive 
zone”.  When  the  third  beam  is  coupled,  the  induced  variation  of  coupling  loss  factor 
depends  on  how  the  amount  of  overlap  between  the  modes  of  subsystems  1  and  2  is 
affected.  It  can  be  increased  or  decreased  and  thus  the  coupling  loss  factor  between 
subsystems  1  and  2  can  also  be  increased  or  decreased  due  to  the  third  coupled 
subsystem.  For  example,  the  amount  of  such  overlap  in  case  (i)  is  decreased  after 
the  third  subsystem  is  coupled.  As  a  result,  the  coupling  loss  factor,  7712,  becomes 
decreased. 

The  above  discussed  variability  of  coupling  loss  factor  due  to  the  third  coupled 
subsystem  has  been  shown  for  individual  cases.  On  the  other  hand,  if  an  ensemble 
of  similar  structures  are  considered,  this  sensitivity  may  be  reduced  (as  it  is  some¬ 
times  positive  or  negative  depending  on  each  special  situation).  However,  such  a 
variability  obtained  from  two  typical  examples  is  nevertheless  very  useful  when 
one  individual  case  is  studied  in  SEA  or  SEA-like  problems.  The  ignorance  of  such 
effect  of  the  other  coupled  subsystems  on  the  coupling  loss  factor  may  become  one 
of  the  possible  error  sources  causing  SEA  failure. 


4  Conclusions 

The  variation  of  coupling  loss  factor  due  to  the  third  coupled  subsystem  is  stud¬ 
ied  in  this  paper.  The  effect  of  a  third  coupled  subsystem  on  the  coupling  loss 
factor  between  the  first  two  coupled  subsystems  depends  on  how  strong  the  third 
subsystem  is  coupled.  Roughly,  along  with  the  damping  in  the  subsystems,  “joint- 
dependent  zone”  and  “modal-sensitive  zone”  are  defined  according  to  the  different 
variation  properties  of  coupling  loss  factor.  In  the  “modal-sensitive  zone”,  the  ef- 


1160 


feet  of  a  third  coupled  subsystem  on  the  coupling  loss  factor  could  be  positive  or 
negative.  It  depends  on  how  the  amount  of  overlap  between  the  modes  of  two  con¬ 
nected  subsystems  is  affected.  This  “modal-sensitive”  effect  may  be  averaged  out 
for  an  ensemble  of  structures,  but  it  is  important  when  SEA  is  applied  to  individual 
cases.  In  the  “joint-dependent  zone”,  the  coupling  loss  factor  is  insensitive  to  the 
strength  of  the  coupling  between  the  second  and  third  subsystems.  Since  the  two 
different  zones  are  allocated  according  to  the  system  damping  (which  is  equivalent 
to  modal  overlap  factor  when  the  central  frequency  and  modal  density  are  fixed), 
it  shows  the  importance  of  reverberance  in  subsystems  when  the  classical  SEA  is 
applied  [7,8].  How  to  take  into  account  the  effect  of  the  other  coupled  subsystems 
in  evaluating  coupling  loss  factor,  especially  when  the  system  damping  is  low  and 
when  an  individual  case  is  considered,  definitely  needs  to  be  further  investigated. 


References 

[  1  ]  Richard  H.  Lyon  and  Richard  G.  DeJong.  Theory  and  Application  of  Statistical 
Energy  Analysis.  Butterworth-Heinemann,  second  edition,  1995. 

[2]  Richard  H.  Lyon  and  G.  Maidanik.  Power  flow  between  linearly  coupled  os¬ 
cillators.  Journal  of  the  Acoustic  Society  of  America,  34:623-639,  1962. 

[3]  Eric  E.  Ungar.  Statistical  energy  analysis  of  vibrating  systems.  Transactions 
of  the  ASME,  Journal  of  Engineering  for  Industry,  pages  626-632,  November 
1967. 

[4]  F.  J.  Fahy.  Statistical  energy  analysis.  In  R.  G.  White  and  J.  G.  Walker,  editors. 
Noise  and  Vibration,  chapter  7,  pages  165-186.  Chichester,  Ellis  Horwood, 
1982. 

[5]  J.  M.  Cuschieri  and  J.  C.  Sun.  Use  of  statistical  energy  analysis  for  rotating 
machinery,  part  II;  Coupling  loss  factors  between  indirectly  coupled  substruc¬ 
tures.  Journal  of  Sound  and  Vibration,  170(2):  191-201,  1994. 

[6]  P.  W.  Smith.  Statistical  models  of  coupled  dynamical  systems  and  the  transi¬ 
tion  from  weak  to  strong  coupling.  Journal  of  the  Acoustic  Society  of  America, 
65:695-698,  1979. 

[7]  S.  Finnveden.  Ensemble  averaged  vibration  energy  flows  in  a  three-element 
structure.  Journal  of  Sound  and  Vibration,  187(3);495-529,  1995. 

[8]  A.  J.  Keane.  A  note  on  modal  summations  and  averaging  methods  as  ap¬ 
plied  to  statistical  energy  analysis  (SEA).  Journal  of  Sound  and  Vibration, 
164(1);  143-156,  1993. 

[9]  B.  R.  Mace.  The  statistical  energy  analysis  of  two  continuous  one-dimensional 
subsystems.  Journal  of  Sound  and  Vibration,  166(3):429-461,  1993. 


1161 


[  10]  Hongbing  Du  and  Fook  Fah  Yap.  A  study  of  damping  effects  on  coupling  loss 
factor  used  in  statistical  energy  analysis.  In  Proceedings  of  the  Fourth  Inter¬ 
national  Congress  on  Sound  and  Vibration,  pages  265-272,  St.  Petersburge, 
Russia,  June  1996. 

[11]  Fook  Fah  Yap  and  J.  Woodhouse.  Investigation  of  damping  effects  on  statis¬ 
tical  energy  analysis  of  coupled  structures.  Journal  of  Sound  and  Vibration, 
197(3):35I-371,  1996. 

[12]  D.  A.  Bies  and  S.  Hamid.  In  situ  determination  of  loss  and  coupling  loss 
factors  by  the  power  injection  method.  Journal  of  Sound  and  Vibration, 
70(2):  187-204,  1980. 

[13]  L.  Cremer,  M.  Heckl,  et  al.  Structure-Borne  Sound:  Structural  Vibrations  and 
Sound  Radiation  at  Audio  Frequencies.  Springer- Verlag,  second  edition,  1987. 

[14]  R.  S.  Langley.  A  derivation  of  the  coupling  loss  factors  used  in  statistical 
energy  analysis.  Journal  of  Sound  and  Vibration,  141(2):207-219,  1990. 

[15]  J.  C.  Sun,  C.  Wang,  et  ai.  Power  flow  between  three  series  coupled  oscillators. 
Journal  of  Sound  and  Vibration,  1 89(2)  :2 15-229,  1996. 


1162 


THE  EFFECT  OF  CURVATURE  UPON  VIBRATIONAL 
POWER  TRANSMISSION  IN  BEAMS 
SJ.  Walsh(l)  and  R.G.White(2) 


(1)  Department  of  Aeronautical  and  Automotive 
Engineering  and  Transport  Studies 
Loughborough  University 

(2)  Department  of  Aeronautics  and  Astronautics 
University  of  Southampton 

ABSTRACT 

Previous  research  into  structural  vibration  transmission  paths 
has  shown  that  it  is  possible  to  predict  vibrational  power 
transmission  in  simple  beam  and  plate  structures.  However,  in 
many  practical  structures  transmission  paths  are  composed  of  more 
complex  curved  elements;  therefore,  there  is  a  need  to  extend 
vibrational  power  transmission  analyses  to  this  class  of  structure. 
In  this  paper,  expressions  are  derived  which  describe  the  vibrational 
power  transmission  due  to  flexural,  extensional  and  shear  types  of 
travelling  wave  in  a  curved  beam  which  has  a  constant  radius  of 
curvature.  By  assuming  sinusoidal  wave  motion,  expressions  are 
developed  which  relate  the  time-averaged  power  transmission  to 
the  travelling  wave  amplitudes.  The  results  of  numerical  studies 
are  presented  which  show  the  effect  upon  power  transmission 
along  a  curved  beam  of:  (i)  the  degree  of  curvature;  and  (ii)  various 
simplifying  assumptions  made  about  the  beam  deformation. 

1.  INTRODUCTION 

Previous  research  into  structural  transmission  paths  has 
shown  it  is  possible  to  predict  vibrational  power  transmission  in 
simple  beam  and  plate  structures.  More  recently,  transmission 
through  pipes  with  bends,  branches  and  discontinuities  has  been 
studied,  which  has  led  to  useful  design  rules  concerning  the 
position  and  size  of  pipe  supports  for  minimum  power 
transmission[l].  However,  in  many  practical  structures 
transmission  paths  are  composed  of  more  complex  curved 
elements.  Therefore,  there  is  a  need  to  extend  power  transmission 
analyses  to  this  class  of  structure. 


1163 


Wave  motion  in  a  curved  beam  with  a  constant  radius  of 
curvature  has  been  considered  by  Love  [2]  who  assumed  that  the 
centre-line  remains  unextended  during  flexural  motion,  whilst 
flexural  behaviour  is  ignored  when  considering  extensional 
motion.  Using  these  assumptions  the  vibrational  behaviour  of 
complete  or  incomplete  rings  has  been  considered  by  many 
researchers  who  are  interested  in  the  low  frequency  behaviour  of 
arches  and  reinforcing  rings.  In  reference  [2]  Love  also  presented 
equations  for  thin  shells  which  include  the  effects  of  extension  of 
the  mid-surface  during  bending  motion.  Soedel  [3]  reduced  these 
equations  and  made  them  applicable  to  a  curved  beam  of  constant 
radius  of  curvature.  In  an  alternative  approach  Graff  [4]  derived 
these  equations  from  first  principles  and  also  constructed  frequency 
verses  wavenumber  and  wavespeed  versus  wavenumber  graphs. 
Philipson  [5]  derived  a  set  of  equations  of  motion  which  included 
extension  of  the  central  line  in  the  flexural  wave  motion,  and  also 
rotary  inertia  effects.  In  a  development  analogous  to  that  of 
Timoshenko  for  straight  beams,  Morley  [6]  introduced  a  correction 
for  radial  shear  when  considering  the  vibration  of  curved  beams. 
Graff  later  presented  frequency  versus  wave  number  and  wave 
speed  versus  wave  number  data  for  wave  motion  in  a  curved  beam, 
when  higher  order  effects  are  included  [7]. 

In  this  paper,  expressions  for  vibrational  power  transmission 
in  a  curved  beam  are  derived  from  first  principles.  In  the  next 
section  two  sets  of  governing  equations  for  wave  motion  in  a 
curved  beam  are  presented  both  of  which  include  coupled 
extensional-flexural  motion.  The  first  set  is  based  upon  a  reduction 
of  Love's  thin  shell  equations  mentioned  above.  The  second  set  is 
based  upon  a  reduction  of  Fliigge’s  thin  shell  equations  [8].  In 
section  three,  the  expressions  for  stresses  and  displacements 
presented  in  section  two  are  used  to  derive  formulae  for  vibrational 
power  transmission  in  terms  of  centre-line  displacements.  By 
assuming  sinusoidal  wave  motion,  expressions  are  developed 
which  relate  the  time-averaged  power  transmission  to  the 
extensional  and  flexural  travelling  wave  amplitudes.  In  section 
four,  corrections  for  rotary  inertia  and  shear  deformation  are 
introduced.  The  results  of  numerical  studies  of  these  expressions 
are  presented  which  show  the  effect  upon  wave  motion  and  power 
transmission  of  (i)  the  degree  of  curvature,  and  (ii)  the  various 
simplifying  assumptions  made  about  the  beam  deformation. 


1164 


2.  WAVE  MOTION  IN  CURVED  BEAMS 


In  this  section  the  governing  relations  between  displacements, 
strains,  stresses  and  force  resultants  in  a  curved  beam  are  presented. 
The  centre-line  of  the  beam  lies  in  a  plane  and  forms  a  constant 
radius  of  curvature.  The  cross-section  of  the  beam  is  uniform  and 
symmetrical  about  the  plane  and  it  is  assumed  that  there  is  no 
motion  perpendicular  to  the  plane.  It  is  also  assumed  that  the  beam 
material  is  linearly  elastic,  homogeneous,  isotropic  and  continuous. 

Consider  a  portion  of  the  curved  beam,  as  shown  in  Figure  1. 
The  circumferential  coordinate  measured  around  the  centre-line  is 
s,  while  the  outward  pointing  normal  coordinate  from  the  centre¬ 
line  is  z,  and  the  general  radial  coordinate  is  r.  A  complete  list  of 
notation  is  given  in  the  appendix.  For  small  displacements  of  thin 
beams  the  assumptions,  known  as  "Love's  first  approximation"  in 
classical  shell  theory,  can  be  made  [8].  This  imposes  the  following 
linear  relationships  between  the  tangential  and  radial 
displacements  of  a  material  point  and  components  of  displacement 
at  the  undeformed  centre-line: 


U  (r,  s,  t)  =  u  (R,  s,  t)  +  z  (|)  (s,  t)  (1) 

W  (r,  s,  t)  =  w  (R,  s,  t)  (2) 

where  u  and  w  are  the  components  of  displacement  at  the  centre¬ 
line  in  the  tangential  and  radial  directions,  respectively,  (j)  is  the 
rotation  of  the  normal  to  the  centre-line  during  deformation: 


(|> 


3w 


/  angle  of'X  { rotational  displacement^ 

Vcurvature/  \  of  straight  beam  ) 


(3) 


and  W  is  independent  of  z  and  is  completely  defined  by  the  centre¬ 
line  component  w. 

Circumferential  strain  consists  of  both  an  extensional  strain 
and  bending  strain  component.  Expressions  for  these  are  listed  in 
table  1.  The  strain-displacement  expressions  of  the  Love  and  Flugge 
based  equations  are  identical.  However,  in  the  total  circumferential 
strain  of  the  Love  based  theory,  the  term  in  the  denominator 
has  been  neglected  with  respect  to  unity.  Assuming  the  material  to 
be  linearly  elastic,  the  circumferential  stress-strain  relationship 
is  given  by  Hooke's  Law,  whilst  the  shear  strain,  Ysr/  and  shear 
stress,  Osr/  are  assumed  to  be  zero.  Assuming  the  material 


1165 


to  be  homogeneous  and  isotropic,  the  material  properties  E,  G  and  v 
can  be  treated  as  constants.  Thus,  by  integrating  the  stresses  over 
the  beam  thickness,  force  and  moment  resultants  can  be  obtained, 
which  are  listed  in  table  2.  The  adopted  sign  convention  is  shown  in 
Figure  2. 

Equations  of  motion  for  a  curved  beam  are  presented  in  [4]. 
These  equations  are  derived  in  terms  of  the  radian  parameter  0.  By 
applying  the  substitution,  s  -  RQ,  the  equations  of  motion  can  be 
expressed  in  terms  of  the  circumferential  length,  s.  These  equations 
are  listed  in  [9]  along  with  the  Fliigge  based  equations  of  motion 
which  have  been  obtained  by  a  reduction  of  the  equations  of 
motion  for  a  circular  cylindrical  shell  presented  in  [8].  An  harmonic 
solution  of  the  equations  of  motion  can  be  obtained  by  assuming 
that  extensional  and  flexural  sinusoidal  waves  propagate  in  the 
circumferential  direction.  The  harmonic  form  of  the  equations  of 
motion  are  also  listed  in  [9]. 


3.  VIBRATIONAL  POWER  TRANSMISSION  IN  CURVED  BEAMS 

In  this  section  the  expressions  for  displacements  and  stresses 
presented  in  section  two  are  used  to  derive  the  structural  intensity 
and  power  transmission  due  to  flexural  and  extensional  travelling 
waves  in  a  curved  beam.  The  structural  intensity  expressions  are 
formulated  in  terms  of  displacements  at  the  centre-line.  By 
assuming  sinusoidal  wave  motion,  expressions  are  developed 
which  relate  the  time-averaged  power  transmission  to  the  flexural 
and  extensional  travelling  wave  amplitudes. 

Structural  Intensity  in  the  circumferential  direction  of  a  curved 
beam  is  given  by  [10]: 


Is  = 


au 
at 

f  intensity  due  to 
I  circumferential  stress 


aw 
at 

f  intensity  due  to 
radial  shear  stress 


(4) 


By  integrating  across  the  beam  thickness  power  transmission  per 
unit  length  in  the  circumferential  direction  is  obtained: 

h/2 

Ps=  J  Isdz  (5) 

-h/2 


1166 


Substituting  the  Love  based  circumferential  stress-strain  relation 
and  strain-displacement  expression  into  equation  (5)  the  power 
transmission  due  to  circumferential  stress  is  obtained.  (A  full 
derivation  is  given  in  [9].)  By  analogy  to  power  transmission  in  a 
straight  beam  [1]  this  can  be  expressed  in  terms  of  an  extensional 
component,  Pe,  and  a  bending  moment  component,  Pbm*  Although 
the  transverse  shear  stress  Ogr  is  negligible  under  Love's  first 
approximation,  the  power  transmission  due  to  transverse  shear 
stress  can  be  evaluated  from  the  non-vanishing  shear  force,  Q, 
because  the  radial  displacement  W  does  not  vary  across  the  beam 
thickness.  Again,  by  analogy  to  power  transmission  in  a  straight 
beam  [1]  this  is  expressed  as  a  shear  force  component.  Thus,  the  total 
power  transmission  in  the  circumferential  direction  is  given  by  the 
sum  of  the  extensional,  bending  moment  and  shear  force 
components.  These  equations  are  listed  in  table  3  along  with  Fliigge 
based  power  transmission  equations  which  are  also  listed  in  table  3. 
Substituting  harmonic  wave  expressions  into  the  Love  and  Fliigge 
based  power  transmission  equations  gives  expressions  for  the  power 
transmission  in  the  circumferential  direction  in  terms  of  travelling 
wave  amplitudes  A  and  B.  For  sinusoidal  wave  motion  it  is  useful 
to  develop  time-averaged  power  transmission  defined  by  [1]: 


T/2 


<Ps> 


--  f 

t“T  J 


Ps  (s,t)  dt 


-  T/2 


(6) 


where  T  is  the  period  of  the  signal.  Time  averaged  Love  and  Fliigge 
based  power  transmission  equations  are  given  in  table  4. 


4.  THE  EFFECT  OF  ROTARY  INERTIA  AND  SHEAR 
DEFORMATION 

It  is  known  that  shear  deformation  and  rotary  inertia  effects 
become  significant  for  straight  beams  as  the  wave  length  approaches 
the  same  size  as  the  thickness  of  the  beam,  and  for  cylindrical  shells 
as  the  shell  radius  decreases  [8].  Thus,  the  objective  in  this  section  is 
to  establish  more  complete  equations  for  power  transmission  in  a 
curved  beam  and  to  show  under  what  conditions  these  specialise  to 
the  simple  bending  equations  presented  in  section  three. 

Rotary  inertia  effects  are  included  by  considering  each  element 
of  the  beam  to  have  rotary  inertia  in  addition  to  translational 
inertia.  Equations  of  motion  for  a  curved  beam  which  include  the 


1167 


effect  of  rotary  inertia  are  presented  in  [7].  These  equations  are 
listed  in  [9]  in  terms  of  the  circumferential  distance  parameter,  s. 
Equations  for  vibrational  power  transmission  can  be  derived  in  the 
same  marmer  as  described  in  section  three.  These  equations  are 
listed  in  tables  3  and  4  where  it  can  be  seen  that  the  extensional  and 
bending  moment  components  when  including  rotary  inertia  effects 
are  identical  to  the  corresponding  Fliigge  based  expressions. 
However,  the  shear  force  component  now  contains  an  additional 
rotary  inertia  term. 

If  shear  deformation  is  included  then  Kirchoff  s  hypothesis  is 
no  longer  valid,  and  the  rotation  of  the  normal  to  the  centre-line 
during  bending,  (|),  is  no  longer  defined  by  equation  (3)  but  is  now 
another  independent  variable  related  to  the  shear  angle,  y. 
However,  unlike  simple  bending  theory,  where  the  transverse 
shear  strain,  Ysr^  is  negligible,  the  transverse  shear  strain  is  now 
related  to  the  shear  angle,  y  which  is  expressed  in  terms  of 
displacements  u,  w  and  (j).  The  circumferential  force,  bending 
moment,  and  shear  force  obtained  from  [8]  are  given  in  table  2.  A  set 
of  equations  of  motion  for  a  curved  beam  which  includes  the  effect 
of  shear  deformation  is  presented  in  [9].  Power  transmission 
equations  in  the  circumferential  direction  can  be  obtained  in  a 
manner  analogous  to  that  used  for  Love  and  Fliigge  based  theories. 
As  before,  the  power  transmission  due  to  circumferential  stress 
can  be  identified  as  consisting  of  extensional  and  bending  moment 
components.  The  contribution  to  the  power  transmission  from  the 
transverse  shear  stress  is  obtained  from  the  product  of  the  shear 
force  resultant  and  the  radial  velocity  which  gives  the  shear  force 
component  of  power  transmission. 


5.  NUMERICAL  STUDY 

For  a  given  real  wavenumber,  k,  the  harmonic  equations  of 
motion  were  solved  to  find  the  corresponding  circular  frequency,  co, 
and  complex  wave  amplitude  ratio.  The  simulated  beam  was 
chosen  to  have  the  physical  dimensions  and  material  properties  of 
typical  mild  steel  beams  used  for  laboratory  experiments.  Four 
different  radii  of  curvature  were  investigated,  which  were 
represented  in  terms  of  the  non-dimensional  thickness  to  radius  of 
curvature  ratio,  h/R.  These  ratios  were  ^/lO/ 100/ 1000  and 
^/lOOOO- 

Using  the  Love  equations  of  motion.  Figure  3  shows  the 
relationship  between  wave  number  and  frequency  for  a  beam  with  a 


1168 


thickness  to  radius  of  curvature  ratio  of  ^/lo-  The  frequency  range 
is  represented  in  terms  of  the  non-dimensional  frequency 
parameter  Q  =  ®^/co,  where  Cq  is  the  phase  velocity  of  extensional 
waves  in  a  straight  bar  and  the  wave  number  range  is  represented 
in  terms  of  the  non-dimensional  wave  number,  kR.  It  can  be  seen 
that  two  types  of  elastic  wave  exist:  one  involving  predominantly 
flexural  motion;  the  other  predominantly  extensional  motion. 
However,  for  wave  numbers  less  than  kR  =  1,  the  predominantly 
flexural  wave  exhibits  greater  extensional  than  flexural  motion. 

Solution  of  the  shear  deformation  equations  of  motion  for  a 
curved  beam  shows  that  three  types  of  elastic  wave  exist.  These  are 
the  predominantly  flexural  and  predominantly  extensional  waves 
of  simple  bending  theory  and  additionally  a  predominantly 
rotational  wave  related  to  the  shear  angle.  The  relationship 
between  wave  number  and  frequency  for  these  three  wave  types  is 
shown  in  Figure  4. 

A  numerical  investigation  of  the  power  transmission 
equations  was  undertaken  using  simulated  beams  with  the  same 
dimensions  and  material  properties  as  those  used  in  the  previous 
study  of  wave  motion.  Figure  5  shows  the  relationship  between 
transmitted  power  ratio  and  frequency.  For  the  predominantly 
flexural  wave  the  time-averaged  transmitted  power  ratio  is 
calculated  by  dividing  the  time-averaged  power  transmitted  along  a 
curved  beam  by  a  predominantly  flexural  wave  by  the  time- 
averaged  power  transmitted  by  a  pure  flexural  wave  travelling  in  a 
straight  Euler-Bernoulli  beam.  i.e.  the  ratio  (<Pe>t  +  <Fbm>t  + 
<Psf>t)/EIcokf3Af .  For  the  predominantly  extensional  wave  the 
transmitted  power  ratio  is  calculated  by  dividing  the  time-averaged 
power  transmitted  along  a  curved  beam  by  a  predominantly 
extensional  wave  by  the  time-averaged  power  transmitted  by  a  pure 
extensional  wave  in  a  straight  rod.  i.e.  the  ratio  (<Pe>t  +  <Pbn>t  + 
<Psf  >t)/EScokex  P^ex 

6.  SUMMARY  AND  CONCLUSIONS 

In  this  paper,  starting  from  first  principles,  expressions  for 
vibrational  power  transmission  in  a  curved  beam  have  been 
derived  using  four  different  theories.  Love’s  generalised  shell 
equations  include  extension  of  the  centre-line  during  bending 
motion  were  the  first  set  of  equations  considered.  Fiiigge's 
equations  also  include  centre-line  extensions  and  were  the  second 
set  of  equations  used.  Corrections  for  rotary  inertia  and  shear 
deformation  produced  the  third  and  fourth  sets  of  governing 


1169 


equations,  respectively.  By  letting  the  radius  of  curvature,  R,  tend 
to  infinity  these  equations  reduce  to  the  corresponding  straight 
beam  expressions  presented  in  [1]. 

Using  the  governing  equations  for  each  theory,  expressions 
were  then  developed  which  related  time-averaged  power 
transmission  to  the  amplitudes  of  the  extensional,  flexural  and 
rotational  displacements.  For  each  theory  the  effects  of  curvature 
upon  the  resulting  wave  motion  and  power  transmission  were 
then  investigated  using  beams  with  different  degrees  of  curvature. 
From  the  results  of  this  study  it  can  be  seen  that  vibrational  power 
transmission  in  curved  beams  can  be  classified  into  three  different 
frequency  regions: 

(i)  below  the  ring  frequency,  Q  =  1,  curvature  effects  are 
important; 

(ii)  above  the  ring  frequency  but  below  the  shear  wave  cut-on 

frequency,  =  1  the  curved  beam  behaves  essentially  as  a 

straight  beam; 

(iii)  above  the  shear  wave  cut-on  frequency,  higher  order  effects 
are  important. 


ACKNOWLEDGEMENT 

The  analytical  work  presented  in  this  paper  was  carried  out  while 

both  authors  were  at  the  Institute  of  Sound  and  Vibration  Research, 

University  of  Southampton.  The  financial  support  of  the  Marine 

Technology  Directorate  Limited  and  the  Science  and  Engineering 

Research  Council  is  gratefully  acknowledged. 

REFERENCES 

1.  J.L.  HORNER  1990  PhD  thesis,  University  of  Southampton 
Vibrational  power  transmission  through  beam  like  structures. 

2.  A.E.H.  LOVE  1940  Dover,  Nezv-York.  A  treatise  on  the 
mathematical  theory  of  elasticity. 

3.  W.  SOEDEL  1985  Dekker,  New  York.  Vibrations  of  shells  and 
plates 

4.  K.F.  GRAFF  1975  Clarenden  Press,  Oxford.  Wave  motion  in 
elastic  solids 

5.  L.L.  PHILIPSON  1956  Journal  of  Applied  Mechanics  23,  364- 
366.  On  the  role  of  extension  in  the  flexural  vibrations  of  rings. 

6.  L.S.D.  MORLEY  1961  Quarterly  Journal  of  Mechanics  and 
Applied  Mathematics  14,  (2),  155-172.  Elastic  waves  in  a 
naturally  curved  rod. 


1170 


7.  K.F.  GRAFF  1970  IEEE  Transactions  on  Sonics  and  Ultrasonics, 
SU-17  (1),  1-6.  Elastic  wave  propagation  in  a  curved  sonic 
transmission  line. 

8.  A.W.  LEISSA  1977  NASA  SP-288,  Washington  DC  Vibrations 
of  shells. 

9.  S.J.  WALSH  1996  PhD  thesis,  University  of  Southampton. 
Vibrational  power  transmission  in  curved  and  stiffened 
structures. 

10.  A.J.  ROMANO,  P.B.  ABRAHAM,  E.G.  WILLIAMS  1990  Journal 
of  the  Acoustical  Society  of  America  87.  A  Poynting  vector 
formulation  for  thin  shells  and  plates,  its  application  to 
structural  intensity  analysis  and  source  localization.  Part  I: 
Theory. 


APPENDIX:  NOTATION 

A  flexural  wave  amplitude 

Af  amplitude  of  a  purely  flexural  wave 

B  extensional  wave  amplitude 

Bex  amplitude  of  a  purely  extensional  wave 

C  rotation  wave  amplitude 

E  Young’s  modulus 

G  shear  modulus 

I  second  moment  of  area  of  cross-section  of  beam 

Ig  structural  intensity  in  circumferential  direction 

K  radius  of  gyration 

M  bending  moment  on  cross-section  of  beam 

N  circumferential  force  on  cross-section  of  beam 

P  transmitted  power 

Q  shear  force  on  cross-section  of  beam 

R  radius  of  curvature 

S  cross-sectional  area  of  beam 

T  period  of  wave 

LF  displacement  in  circumferential  direction 

W  displacement  in  radial  direction 

Co  wavespeed  of  extensional  waves  in  a  straight  bar 

Cs  wavespeed  of  shear  waves  in  a  straight  bar 

ds  length  of  elemental  slice  of  curved  beam 

eg  total  circumferential  strain 

h  thickness  of  beam 

k  wavenumber 

kex  wavenumber  of  a  purely  extensional  wave 

kf  wavenumber  of  a  purely  flexural  wave 

r  coordinate  in  radial  direction 


1171 


s  coordinate  in  circumferential  direction 

u  displacement  at  centre-line  in  circumferential  direction 

w  displacement  at  centre-line  in  radial  direction 

z  coordinate  of  outward  pointing  normal 

Q.  non-dimensional  frequency 

Ps  bending  strain 

Y  shear  angle 

Ysr  transverse  shear  strain 

Ej.  radial  strain 

£g  circumferential  strain 

0A  phase  angle  of  flexural  wave 

0C  phase  angle  of  rotational  wave 

K  Timoshenko  shear  coefficient 

?iex  wave  length  of  extensional  waves  in  a  straight  bar 

V  Poisson's  ratio 

Oj.  radial  stress 

Gg  circumferential  stress 

0SJ  transverse  shear  stress 

d  change  in  slope  of  normal  to  centre-line 

CO  radian  frequency 


W(r,s,t) 


1172 


Figure  2:  Sign  convention  and  force  resultants  on  an  elemental 
slice  of  curved  beam 


Figure  3:  Wave  number  v.  frequency  relationship  for  a  curved 
beam  predicted  using  Love  theory 


Transmitted  power  ratio  (Curved  beam)/(Slraight  beam)  ^  Non-dimensional  wavenumber 


Table  1:  Displacement,  strain-displacement  and  stress-strain  equations  for  a  curved  beam. 


1175 


Table  2:  Force  resultants  for  a  curved  beam. 


1176 


Table  3:  Power  transmission  for  a  curved  beam. 


1177 


Table  4:  Time-averaged  power  transmission  by  a  single  harmonic  wave 


1178 


A  Parameter-based  Statistical  Energy 
Method  for  Mid-frequency 
Vibration  Transmission  Analysis 

Sungbae  Choi,  Graduate  Student  Research  Assistant 
Matthew  P.  Castanier,  Assistant  Research  Scientist 
Christophe  Pierre,  Associate  Professor 
Department  of  Mechanical  Engineering  and  Applied  Mechanics 
The  University  of  Michigan 
Ann  Arbor,  MI  48109-2125,  USA 

Abstract 

Vibration  transmission  between  two  multi-mode  substructures  con¬ 
nected  by  a  spring  is  investigated.  A  classical  Statistical  Energy 
Analysis  (SEA)  approach  is  reviewed,  and  it  is  seen  that  some  typ¬ 
ical  assumptions  which  are  valid  at  high  frequencies  lose  accuracy 
in  the  mid-frequency  range.  One  assumption  considered  here  is 
that  of  an  identical  probability  density  function  (pdf)  for  each  reso¬ 
nant  frequency.  This  study  proposes  a  Parameter-based  Statistical 
Energy  Method  (PSEM)  which  considers  individual  modal  informa¬ 
tion.  The  results  of  PSEM  have  good  agreement  with  those  of  a 
Monte  Carlo  technique  for  an  example  system. 

Nomenclature 


E[  ] 

expected  value 

power  transmitted  between  substructure  i  and  k 

IIfc(u;)  . 

total  power  transmitted  to  substructure  k 

Pi 

power  input  to  substructure  i 

Vi, 

coupling  loss  factor  (CLF) 

fh 

modal  driving  force  for  mode  j  of  substructure  1 

UJ 

frequency  [rad/sec] 

l.l.-.lj.Ol 

subscripts  for  decoupled  Bar  1 

2,2r,2„02 

subscripts  for  decoupled  Bar  2 

El,  ^2 

blocked  energy 

Eoi ,  Eo2 

Young’s  modulus 

Pl^P2 

density 

mi, m2 

mass  per  unit  length 

Mu  M2 

total  mass 

Cl,C2 

viscous  damping  ratio 

Ai,  A2 

cross-sectional  area 

0 

rH 

0 

nominal  length 

1179 


Ti,T2 

disordered  length 

£1,^2 

ratio  of  disorder  to  nominal  length 

k 

coupling  stiffness 

coupling  ratio 

Xi,X2 

position  coordinate 

ai,a2 

point- coupling  connection  position 

Wi,  W2 

deflection 

modal  amplitude 

mode  shape  function 

NuN2 

number  of  resonant  frequencies 

resonant  frequencies 

lower  limit  of  resonant  frequencies 

upper  limit  of  resonant  frequencies 

(Tl,  (72 

standard  deviation  of  disorder 

1  Introduction 

Vibration  transmission  analysis  between  connected  substructures  in  the 
mid-frequency  range  is  often  a  daunting  prospect.  Since  the  analysis  at 
high  frequencies  requires  greater  model  discretization,  the  size  and  com¬ 
putational  cost  of  a  full  structure  model  (e.g.,  a  Finite  Element  model) 
can  become  prohibitive.  Also,  as  the  wavelengths  approach  the  scale  of 
the  structural  variations,  uncertainties  (tolerances,  defects,  etc.)  can  sig¬ 
nificantly  affect  the  dynamics  of  the  structure.  Starting  at  what  may  be 
called  the  mid-frequency  range,  deterministic  models  fail  to  predict  the 
response  of  a  representative  structure  with  uncertainties. 

Therefore,  in  the  mid-frequency  range,  a  statistical  analysis  of  vibra¬ 
tion  transmission  may  be  more  appropriate.  This  approach  is  taken  in 
the  procedure  known  as  Statistical  Energy  Analysis  (SEA)  [1].  In  SEA, 
a  structure  is  divided  into  coupled  substructures.  It  is  assumed  that  each 
substructure  exhibits  strong  modal  overlap  which  makes  it  difficult  to  dis¬ 
tinguish  individual  resonances.  Therefore,  the  resonant  frequencies  are 
treated  as  random  variables,  each  with  an  identical,  uniform  probability 
density  function  (pdf)  in  the  frequency  range  of  interest.  This  assumption 
greatly  simplifies  the  evaluation  of  the  expected  value  of  transmitted  vi¬ 
bration  energy.  A  simple  linear  relation  of  vibration  transmission  between 
each  pair  of  substructures  is  retrieved.  The  power  transmitted  is  propor¬ 
tional  to  the  difference  in  the  average  modal  energies  of  the  substructures. 
This  relation  is  analogous  to  Fourier’s  law  of  heat  transfer  [1-4]. 

In  the  low-  to  mid-frequency  region,  the  modal  responses  are  not  strongly 
overlapped.  In  this  case,  two  typical  SEA  assumptions  are  less  accurate:  an 
identical  pdf  for  all  resonant  frequencies,  and  identical  (ensemble-averaged) 


1180 


values  of  the  associated  mode  shape  functions  at  connection  positions.  In 
this  paper,  these  two  assumptions  are  relaxed.  A  distinct  uniform  pdf 
is  applied  for  each  resonant  frequency,  and  a  piecewise  evaluation  of  the 
transmitted  power  is  performed.  This  is  called  a  Parameter-based  Sta¬ 
tistical  Energy  Method  (PSEM)  because  it  considers  the  statistical  char¬ 
acteristics  of  individual  system  parameters.  This  solution  can  accurately 
capture  peaks  of  transmitted  power  while  maintaining  the  SEA  advantage 
of  efficiency. 

This  paper  is  organized  as  follows.  In  section  2,  we  briefly  review  SEA 
along  with  the  associated  assumptions  and  limitations.  In  section  3,  the 
power  transmitted  between  two  spring- coupled  multi-mode  substructures 
is  investigated  by  applying  several  SEA  assumptions.  A  Monte  Carlo  solu¬ 
tion  is  used  for  comparison.  In  section  4,  the  PSEM  approach  is  presented 
and  the  results  are  shown.  Finally,  section  5  draws  conclusions  from  this 
study. 

2  Overview  of  SEA 

In  Statistical  Energy  Analysis,  the  primary  variable  is  the  time-averaged 
total  energy  of  each  substructure.  This  is  called  the  blocked  energy,  where 
blocked  means  an  assumed  coupling  condition.  The  assumed  coupling  con¬ 
dition  may  be  the  actual  coupling,  a  clamped  condition  at  the  substructure 
junctions,  or  a  decoupled  condition  [2,  5]. 

In  order  to  predict  the  average  power  transmitted  between  two  directly- 
coupled  substructures,  a  few  simplifying  assumptions  are  applied.  Some  of 
the  essential  SEA  assumptions  are  summarized  by  Hodges  and  Woodhouse 
in  Ref.  [3]; 

•  Modal  incoherence:  the  responses  of  two  different  modal  coordinates 
are  uncorrelated  over  a  long  time  interval 

•  Equipartition  of  modal  energy:  all  modes  within  the  system  have  the 
same  kinetic  energy 

The  above  conditions  make  it  possible  to  treat  all  modal  responses  as  sta¬ 
tistically  identical.  The  first  assumption  implies  a  broad  band,  distributed 
driving  force  (often  called  ”rain  on  the  roof’)  which  leads  to  uncorrelated 
modal  driving  forces.  The  second  assumption  implies  that  the  substruc¬ 
tures  have  strong  modal  overlap,  or  that  the  parameter  uncertainties  are 
sufficiently  large  that  the  modes  are  equally  excited  in  an  ensemble  average 
sense.  Thus,  the  resonant  frequencies  are  treated  as  random  variables  with 
identical,  uniform  probability  density  functions  (pdfs)  for  the  frequency 
range  of  interest. 

The  SEA  relation  for  the  expected  value  of  power  transmitted  from 


1181 


substructure  i  to  substructure  k,  may  be  expressed  as 

E[nijfc(cj)]  =  0)7}, ^Ni{^  (1) 

where  uj  is  the  frequency,  77^^  is  the  coupling  loss  factor,  Ei  is  the  blocked 
energy  of  substructure  z,  and  Ni  is  the  number  of  participating  modes  of 
substructure  i  for  the  frequency  range  of  interest.  The  power  dissipated  by 
substructure  i  is  expressed  as 

=  u:riiEi  (2) 

where  r]i  is  the  damping  loss  factor.  Using  Eqs.(l)  and  (2),  the  equation 
of  power  balance  for  substructure  i  at  steady  state  [1,  2,  5,  6]  is 

N  N 

Pi  =  E[Ilij]  +  UJTJiEi  =  UJ  (3) 

i=i 

j¥i 

where  Pi  is  the  power  input  to  substructure  i  from  external  sources.  Note 
that  the  first  term  on  the  right-hand  side  is  the  power  transmitted  through 
direct  coupling  between  substructures. 


3  Vibration  Transmission  in  a  Two-Bar 

System 

The  longitudinal  vibration  of  the  structure  shown  in  Fig.l  is  considered 
in  this  study.  The  structure  consists  of  two  uniform  bars  with  viscous 
damping  which  are  coupled  by  a  linear  spring  of  stiffness  k.  The  spring  is 
connected  at  intermediate  points  on  the  bars,  Xi  =  ai  and  X2  =  0,2.  Bar 
i  has  nominal  length  Loi.  A  parameter  uncertainty  may  be  introduced  by 
allowing  the  length  to  vary  by  a  small  random  factor  e^,  which  is  referred 
to  as  disorder.  The  length  of  a  disordered  bar  is  Li  =  Zrox(l  +  £:)•  The  ratio 
of  the  connection  position  to  the  length,  aifLi^  is  held  constant.  Bar  1  is 
excited  by  a  distributed  force  Fi{xi,t). 

3.1  Nominal  transmitted  power 

The  power  transmitted  from  Bar  1  to  Bar  2  for  the  nominal  system  (no 
disorder)  is  briefly  presented  here.  A  more  detailed  derivation  is  shown  in 
the  Appendix  (see  also  Refs.  [7]  and  [8]).  The  equations  of  motion  are 

Wi(a:i,t)  -f  ^Ci(wi(a;i,f))  =  Fi(a:i,t)-|- 

^[W2(a2,  t)-wi(ai,  -  ci) 

(jBo2A2-§^  +  m2|^^W2(x2,t)  +  ^C'2(w2(a;2,t))  = 

k[wi{ai,t)-W2{a2,t)]6{x2  -  ^2) 


1182 


a) 

Bar  2 


Fig.  1:  Two-bar  system 

where  ^  is  a  Dirac  delta  function,  and  (for  Bar  i)  Eoi  is  Young’s  modulus, 
Ai  is  the  cross-sectional  area,  rm  is  the  mass  per  unit  length,  ’Ci  is  the 
viscous  damping  operator,  and  Wt(a:i,t)  is  the  deflection.  The  deflections 
of  the  two  bars  can  be  expressed  by  a  summation  of  modes: 

Wi{xi,t)  =  W2(x2,t)  =  E  1^2r(i)^2r(^2)  (5) 

t=0  r=0 

where  and  W2^{t)  are  modal  amplitudes,  and  and  ^2r(^2) 

are  mode  shape  functions  of  the  decoupled  bars.  These  mode  shape  func¬ 
tions  are  normalized  so  that  each  modal  mass  is  equal  to  the  total  mass  of 
the  bar,  Mi.  Applying  modal  analysis  and  taking  a  Fourier  transform,  the 
following  equations  are  obtained: 

Mi(f)ijWij  =  4-  fc^ij(ai)[E  kF2r^2r(<^2)  -  E 

_  (6) 

M2<f>2,W2,  =  ^^^2,(<22)[E  VFi,^i,(ai)  -  E  W2.^2M2)] 

i=0  r=0 

[ul.  -  -f  •  2Cia;i  .a;)(2  -  sgn(j)) 
y/^  •  2C2<^2.^)(2  -  sgn(s)) 

1  for  ^  >  0 
0  for  2  =  0 

where  an  over-bar  (")  denotes  a  Fourier  transform,  and  are  resonant 
frequencies,  (i  and  (2  are  damping  ratios,  and  is  a  modal  driving  force. 
Mode  0  is  a  rigid  body  mode,  which  is  why  the  sgn(i)  term  is  present. 
Note  that  the  damping  ratio  of  each  bar  is  assumed  to  be  the  same  for  all 
modes. 

Next,  the  modal  driving  forces  are  assumed  to  be  incoherent,  and  each 
spectral  density  function  is  assumed  to  be  constant  (white  noise)  over  a 
finite  range  of  frequency  [2].  After  some  algebra,  the  power  transmitted 


<^2^  = 

sgn(0  = 


1183 


(7) 


from  Bar  1  to  Bar  2,  1112(0;),  is  found  as: 

2o;^PC2«^PlPl  ^  ^2r(^2)0^2r 

M1W2IAI2  ^  |(^2.|2 

-^1  -^2  <?^2r 

where  5pjpi  is  the  same  uniform  spectral  density  function  for  each  modal 
driving  force  on  Bar  1. 


1112(0;)  = 
A  = 


3.2  Monte  Carlo  Energy  Method  (MCEM) 

The  disordered  case  is  now  considered,  where  each  bar  has  a  random  length. 
The  ensemble-averaged  transmitted  power  for  a  population  of  disordered 
two-bar  systems  is  found  by  tahing  the  expected  value  of  Eq.  (7): 


^[ni2] 


(8) 


where  Ni  is  the  number  of  modes  taken  for  Bar  i  (this  is  an  arbitrary  set  of 
modes  that  have  been  aliased  to  the  numbers  1,2,-  •  ’,Ni).  Since  a  truncated 
set  of  modes  is  used,  Eq.  (8)  is  an  approximation.  The  random  variables 
in  Eq.  (8)  are  the  resonant  frequencies  of  the  bars  (which  are  present  in 
the  terms  ^i;,  ^2r?  s,nd  A). 

Equation  (8)  may  be  solved  numerically  using  a  Monte  Carlo  method: 
the  random  variables  are  assigned  with  a  pseudo-random  number  generator 
for  each  realization  of  a  disordered  system,  and  the  transmitted  power 
is  averaged  for  many  realizations.  This  is  called  a  Monte  Carlo  Energy 
Method  (MCEM)  here.  It  may  be  used  as  a  benchmark  for  comparing  the 
accuracy  of  other  approximate  methods. 

Note  that  the  resonant  frequencies  of  a  bar  may  be  found  directly  from 
the  disordered  length.  Therefore,  for  the  MCEM  results  in  this  study,  the 
actual  number  of  random  variables  in  Eq.  (8)  is  taken  to  be  one  for  each 
decoupled  bar.  That  is,  the  two  random  lengths  are  assigned  for  each  real¬ 
ization,  and  then  the  natural  frequencies  are  found  for  each  bar  in  order  to 
calculate  the  transmitted  power.  If  such  a  relation  were  not  known,  each 
resonant  frequency  could  be  treated  as  an  independent  random  variable. 


3.3  SEA-equivalent  Transmitted  Power 

An  SEA  approximation  of  the  transmitted  power  may  be  obtained  by  ap¬ 
plying  several  typical  SEA  assumptions  to  Eq.  (8).  (Since  Eq.  (1)  is  not 
used  directly,  this  might  be  called  an  SEA-equivalent  transmitted  power.) 
These  assumptions  were  summarized  in  Ref.  [8]:  the  coupling  between 


1184 


substructures  is  weak,  the  modal  responses  are  uncorrelated,  the  expected 
value  of  the  square  of  mode  shape  functions  at  connection  positions  is  unity, 
and  the  pdfs  of  the  resonant  frequencies  are  uniform  and  identical. 

The  assumption  of  weak  coupling  means  that  the  value  of  jAj  in  Eq. 
(8)  is  approximately  one.  Applying  the  second  and  third  assumptions  then 
yields 


E[ai2] 


M?M2 


(9) 


Since  the  pdf  of  each  resonant  frequency  is  assumed  to  be  uniform,  the 
expected  values  in  Eq.  (9)  are 


E[ 


E[ 


l/4u)" 

1 

.2cos| 

-f  2a;a;i.  •  cosf  + 
^  ul.  —  2u}ujii  •  cos|  • 

•  sin  I 


sin% 


ian~ 


U) 


2  _ 


(10) 


l/4a;"C2Vr^,  -  1) 

- 1 - tan  — — - - — 


W2r=‘*'2ru 


—  ^2r 


4w^C2\/i  -  Cl 


(11) 


W2r=‘*'2rj 


where  a  =  cos~^(l  -  2Cj),  subject  to  the  restrictions  (1  -  2(1)^  <  1  and 
(1  _  2(2)2  ^  I  Finally,  since  the  pdfs  of  the  resonant  frequencies  are  taken 
to  be  identical,  the  frequency  limits  do  not  depend  on  the  individual  modes 
(a;i.  =  uj2r,  -  a^nd  =  u;2,„  =  oju)-  Therefore,  each  sum  in  Eq.  (9) 
simplifies  to  the  product  of  the  expected  value  and  the  number  of  modes 
in  the  frequency  range  of  interest: 


E[n^2] 


2C2A:ViViiV25,^p^  r.r  1  1  Pr  ^2.  ^ 

WM2  Vi.r 


(12) 


Equation  (12)  is  the  SEA  approximation  used  in  this  study. 


3.4  Example 

The  three  formulations  of  the  transmitted  power  presented  thus  far  the 
nominal  transmitted  power  in  Eq.  (7),  the  MCEM  transmitted  power  in 
Eq.  (8),  and  the  SEA-equivalent  transmitted  power  in  Eq.  (12)  —  are 
now  compared  for  a  two-bar  system  with  the  parameters  shown  in  Table  1. 
For  the  MCEM  results,  the  disorder  (ei  and  62)  was  taken  to  be  uniformly 
distributed  with  mean  zero  and  standard  deviation  ai  =  0-2  =  10%. 

As  a  measure  of  the  coupling  strength,  the  coupling  ratio,  Ri,  is  defined 
cLS  the  ratio  of  coupling  stiffness  to  the  equivalent  stiffness  of  a  bar  at  the 


1185 


Table  1:  Material  properties  and  dimensions  of  two  bars 


MIIM2 

21.53/21,53 

[Kg] 

R01IL02 

10.58/8.817 

[m] 

E01IR02 

200  XIOV2OO  xlO^ 

[N/m'1 

ai/a2 

2.116/7.053 

N 

Pllp2 

7,800/7,800 

[Kg/m'*l 

^pipi 

1 

[N^l 

C1/C2 

0.005/0.005 

k 

4.868  xlO^ 

[N/m; 

Fig.  2:  Comparison  of  the  nominal  transmitted  power,  the  MCEM  results 
(20,000  realizations  with  (j^  =  <72-  10%),  and  the  SEA  approximation. 


fundamental  resonant  frequency,  Ri  =  weak  cou¬ 

pling  is  considered  here  such  that  Ri  =  0.01. 

The  nominal  transmitted  power,  the  MCEM  results,  and  the  SEA  ap¬ 
proximation  are  shown  in  Fig.2.  The  transmitted  power  calculated  for 
the  nominal  system  exhibits  distinct  resonances.  This  is  due  to  the  low 
modal  overlap  of  the  bars  in  this  frequency  range.  The  MCEM  results 
show  distinct  peaks  for  uj  <  15,000  rad/s,  but  they  become  smooth  as 
the  frequency  increases.  The  SEA  approximation  does  not  capture  indi¬ 
vidual  resonances.  However,  at  the  higher  frequencies  where  the  disorder 
effects  are  stronger,  the  SEA  approximation  agrees  well  with  the  MCEM 
results.  The  frequency  range  between  where  the  MCEM  results  are  close 
to  the  nominal  results  and  where  they  are  close  to  the  SEA  results  (ap¬ 
proximately  2,500  ~  15,000  rad/s  for  this  case)  is  considered  to  be  the 
mid-frequency  range  here.  This  range  will  vary  depending  on  the  system 


1186 


parameters  and  the  disorder  strength.  In  the  next  section,  an  efficient  ap¬ 
proximation  of  the  transmitted  power  is  presented  which  compares  well 
with  MCEM  in  the  mid-frequency  range. 


4  Parameter-based  Statistical  Energy 
Method  (PSEM) 

The  SEA  approximation  presented  in  the  previous  section  does  not  capture 
the  resonances  in  the  transmitted  power  because  of  two  assumptions:  the 
resonant  frequencies  all  have  the  same  uniform  pdf,  and  the  values  of  the 
square  of  mode  shape  functions  at  the  connection  positions  are  taken  to  be 
the  ensemble- averaged  value.  Keane  proposed  an  alternate  pdf  of  resonant 
frequencies  in  order  to  apply  SEA  to  the  case  of  two  coupled  nearly  periodic 
structures  [9].  This  pdf  is  shown  in  Fig.3(a).  It  accounts  for  the  fact  that 


Fig.  3:  (a)  The  pdf  of  the  natural  frequencies  and  the  resultant  transmitted 
power  from  Ref.  [9].  (b)  The  pdfs  of  three  natural  frequencies,  and  a 
schematic  representation  of  the  piecewise  evaluation  of  transmitted  power 

for  PSEM.  The  individual  modal  contributions  are  extrapolated  ( - )  and 

summed  to  calculate  the  total  transmitted  power  ( — ). 

the  natural  frequencies  of  a  nearly  periodic  structure  tend  to  be  grouped  in 
several  distinct  frequency  bands.  Thus  the  pdf  has  a  large  constant  value 
for  those  frequency  bands,  and  a  small  constant  value  elsewhere.  The  SEA 
approximation  of  transmitted  power  is  then  modified  by  simply  adding  a 
positive  value  or  negative  value  on  a  logarithmic  scale,  as  demonstrated  in 
Fig,  3(a).  This  solution  thus  captures  some  of  the  resonant  behavior  of  the 
transmitted  power. 


1187 


Here,  a  more  general  approach  is  taken  for  approximating  the  power 
transmitted  between  two  substructures  in  a  frequency  range  in  which  they 
have  low  or  intermediate  modal  overlap.  Each  resonant  frequency  is  as¬ 
signed  a  uniform  pdf.  However,  the  frequency  range  of  each  pdf  is  diiferent; 
it  corresponds  to  the  range  in  which  that  resonant  frequency  is  most  likely 
to  be  found.  (The  concept  of  using  “confidence  bands”  as  one-dimensional 
pdfs  was  suggested  but  not  pursued  in  Ref.  [8].)  An  example  is  shown  in 
Fig.  3(b)  for  three  resonant  frequencies.  Furthermore,  it  is  assumed  that 
the  values  of  the  square  of  the  mode  shape  functions  at  the  connection 
positions  are  known.  Thus,  applying  only  the  first  two  SEA  assumptions 
along  with  those  noted  above,  Eq.  (8)  becomes: 


N2 

i:kme 


This  is  called  a  Parameter-ba.sed  Statistical  Energy  Method  (PSEM)  be¬ 
cause  it  employs  information  for  individual  modal  parameters. 

Since  each  modal  pdf  is  uniform,  Eqs.  (10)  and  (11)  still  hold  for  the 


expected  values  in  Eq.  (13).  However,  unlike  the  SEA  approximation,  each 
expected  value  is  different,  because  the  corresponding  frequency  bounds  are 
unique.  Furthermore,  note  that  the  pdfs  do  not  cover  the  entire  frequency 
range  of  interest.  The  results  for  each  mode  are  therefore  extrapolated 
outside  the  frequency  range  of  that  modal  pdf  before  the  individual  modal 
contributions  are  summed.  This  is  shown  schematically  in  Fig.  3(b).  PSEM 


is  therefore  a  piecewise  evaluation  of  the  expected  value  of  transmitted 


power. 

The  PSEM  approximation  is  now  applied  to  the  two-bar  system  of 
Table  1,  with  the  standard  deviation  of  disorder  cti  =  0-2  =  10%.  The  pdfs 
of  the  resonant  frequencies  of  Bar  1  and  Bar  2  aze  shown  in  Fig.  4(a)  and 
(b),  respectively.  For  this  system,  the  bounds  for  each  resonant  frequency 
may  be  found  directly  from  the  variation  of  the  uncertain  parameter.  It 
can  be  seen  that  the  spread  of  each  natural  frequency  pdf  due  to  disorder 
increases  with  increasing  frequency. 

The  MCEM,  PSEM,  and  SEA  approximations  for  the  transmitted  power 
are  shown  in  Fig.  4(c).  There  were  20,000  realizations  taken  for  the  MCEM 
results  at  each  sampled  frequency.  This  took  about  10  hours  of  computa¬ 
tion  time.  In  contrast,  the  PSEM  results  only  required  3  seconds  of  compu¬ 
tation  time,  and  the  SEA  results  only  required  about  1  second.  Note  that 
the  PSEM  results  show  excellent  agreement  with  the  much  more  expensive 
MCEM  results.  The  difference  at  very  low  frequencies  comes  from  the  fact 
that  for  the  PSEM  approximation,  the  value  of  the  term  |A|  was  assumed 
to  be  one  due  to  weak  coupling.  This  assumption  breaks  down  as  the 
frequency  approaches  zero.  However,  the  match  between  the  MCEM  and 


1188 


Fig.  4:  (a)  Natural  frequency  pdfs  for  Bar  1.  (b)  Natural  frequency  pdfs 
for  Bar  2.  (c)  Transmitted  powers  obtained  by  MCEM,  PSEM,  and  SEA 
for  c7i  =  <J2  =  10%. 

PSEM  results  in  the  mid-frequency  range  is  excellent.  Again,  it  is  noted 
that  the  SEA  results  converge  to  those  of  MCEM  (and  PSEM)  as  the  fre¬ 
quency  increases.  Now  it  can  be  seen  that  the  assumption  of  identical  pdfs 
for  all  modes  becomes  better  with  increasing  frequency. 

Next,  the  example  system  is  considered  with  smaller  disorder,  Ci  = 
(72  =  1%.  Fig.  5  shows  the  results  for  this  case  for  what  might  be  called 
the  mid-frequency  range.  Note  that  even  though  this  is  a  higher  frequency 
range  than  that  considered  for  the  previous  ca.se,  the  pdfs  of  the  resonant 
frequencies  shown  in  Fig.  5(a)  and  (b)  are  not  as  strongly  overlapped. 
Thus,  several  peaks  are  seen  in  the  transmitted  power  in  Fig.  5(c).  Again, 
the  PSEM  approximation  agrees  well  with  the  MCEM  results,  although 
there  is  more  discrepancy  for  this  case.  The  SEA  approximation  follows 
the  global  trend,  but  does  not  capture  the  resonances  or  anti-resonances. 
The  SEA  results  drop  off  at  the  edges  because  only  modes  within  this 
frequency  range  are  considered  to  contribute  to  the  transmitted  power. 

In  addition  to  PSEM,  another  piecewise  evaluation  of  the  transmitted 
power  is  considered  here.  For  this  approximation,  wherever  the  individual 
mode  pdfs  overlap,  they  are  superposed  to  form  a  pdf  for  all  the  modes  in 
that  “section”  of  the  frequency  range.  This  superposition  is  demonstrated 
in  Fig.  6.  Also,  if  the  number  of  modes  in  a  section  is  above  a  certain  cutoff 
number,  Nc,  then  it  is  assumed  that  their  mode  shape  function  values  at 
the  connection  positions  are  unknown,  so  that  the  ensemble-averaged  value 
must  be  used.  This  is  called  a  multiple  mode  approximation.  The  purpose 


1189 


Fig.  5:  (a)  Natural  frequency  pdfs  for  Bar  1.  (b)  Natural  frequency  pdfs 
for  Bar  2.  (c)  Transmitted  power  obtained  by  MCEM,  PSEM,-and  SEA 
for  cTi  =  0-2  =  1%. 


Fig.  6:  Resonant  frequency  pdfs  for  PSEM  and  for  the  multiple  mode 
approximation. 

of  formulating  this  approach  is  to  investigate  what  happens  as  information 
about  the  individual  modes  is  lost. 

The  multiple  mode  approximation  is  applied  to  the  example  system 
with  cTi  =  <72  =  10%  in  Fig. 7.  For  A/c  =  2,  this  approximation  has  good 
agreement  with  MCEM.  The  match  is  especially  good  for  u)  <  10,000. 
Above  this  frequency,  the  number  of  overlapped  resonant  frequencies  in 
each  pdf  section  is  greater  than  Nc,  and  the  loss  of  mode  shape  information 
affects  the  results  slightly.  For  Nc  =  0,  the  values  of  the  mode  shape 
functions  are  taken  to  be  one  for  the  entire  frequency  range,  just  as  in  the 
SEA  approximation.  As  can  be  seen  in  Fig.  7,  the  piecewise  construction 
of  the  pdf  roughly  captures  the  frequency  ranges  of  the  resonances  and 
anti-resonances.  However,  the  mode  shape  effect  is  more  pronounced  in 


1190 


-B.S 


-  MCEM 

-e.s  —  Multiple  mode  approximation  rvt=  2 

.  Multiple  mode  approximation  Nfc=  O 


Fig.  7:  Transmitted  power  obtained  from  MCEM  and  the  multiple  mode 
approximation  for  Ui  =  <72  =  10% 

the  mid-frequency  range.  The  peak  values  are  now  similar  to  the  SEA 
approximation. 


5  Conclusions 

In  this  study,  the  power  transmitted  between  two  multi-mode  substruc¬ 
tures  coupled  by  a  spring  was  considered.  A  Monte  Carlo  Energy  Method 
(MCEM)  was  used  to  calculate  the  ensemble  average  of  the  transmitted 
power  for  the  system  with  parameter  uncertainties.  A  classical  Statistical 
Energy  Analysis  (SEA)  approximation  matched  the  Monte  Carlo  results 
in  the  high-frequency  range,  but  did  not  capture  the  resonant  behavior  of 
the  transmitted  power  in  the  mid-frequency  range  where  the  substructures 
have  weak  modal  overlap. 

A  Parameter-based  Statistical  Energy  Method  (PSEM)  was  presented 
which  uses  a  distinct  pdf  for  each  natural  frequency  as  well  as  some  indi¬ 
vidual  mode  shape  information.  A  piecewise  evaluation  of  the  transmitted 
power  was  performed,  and  then  the  modal  contributions  were  extrapolated 
and  superposed.  The  PSEM  approximation  compared  very  well  with  the 
much  more  expensive  Monte  Carlo  results,  including  in  the  mid-frequency 


1191 


range. 


References 


1  R.  H.  Lyon.  Statistical  Energy  Analysis  of  Dynamical  Systems:  Theory 
and  Applications.  M.I.T.  Press,  1st  edition,  1975. 

2  R.  H.  Lyon.  Theory  and  Application  of  Statistical  Energy  Analysis. 
Butterworth-Heinemann,  2nd  edition,  1995. 

3  C.  H.  Hodges  and  J.  Woodhouse.  Theories  of  noise  and  vibration  trans¬ 
mission  in  complex  structures.  Rep. Prog. Physics,  49:107-170,  1986. 

4  J.  Woodhouse.  An  approach  to  the  theoretical  background- of  statis¬ 
tical  energy  analysis  applied  to  structural  vibration.  Journal  of  the 
Acoustical  Society  of  America,  69(6):1695-1709,  1981. 

5  M.  P.  Norton.  Fundamentals  of  Noise  and  Vibration  Analysis  for  En¬ 
gineers.  Cambridge  University  Press,  1st  edition,  1989. 

6  N.  Lalor.  Statistical  energy  analysis  and  its  use  as  an  nvh  analysis  tool. 
Sound  and  Vibration,  30(l):16-20,  1996. 

7  Huw  G.  Davies.  Power  flow  between  two  coupled  beams.  Journal  of 
the  Acoustical  Society  of  America,  51(1):393-401,  1972. 

8  A.  J.  Keane  and  W.  G.  Price.  Statistical  energy  analysis  of  strongly 
coupled  systems.  Journal  of  Sound  and  Vibration,  117(2):363-386, 
1987. 

9  A.  J.  Keane.  Statistical  Energy  Analysis  of  Engineering  Structures 
(Ph.D  Dissertation).  Brunei  University  (England),  1988. 


Appendix 


In  this  appendix,  the  nominal  transmitted  power  in  Eq.  (7)  for  the  mono- 
coupled  two-bar  system  is  derived.  The  procedure  follows  that  of  Refs  [7, 
8]. 

Plugging  Eq.  (5)  into  Eq.  (4), 


F^ixut)  +  fcf  E  -  Oi)  (A.1) 


t=0 


1192 


k  £  Wi,(<)$i,(a,)  -  £  lV2,(«)«2,(a2)  5(x2  -  aj) 


Multiplying  Eq.  (A.l)  by  and  integrating  with  respect  to  xi  for 

[0,  Li]  yields 


MiK  +  = 

A,  +  f f;  W2,(4)W2Xo2)  -  E 


Li 

2CijUijMi  =  y  C'i(«'i,.(a:i))Wij(xi)cia:i  ,  fi,  J  Fi{xi,t)^i.{xi)dx], 

0  0 

and  wij  is  the  ;th  resonant  frequency  of  decoupled  Bar  1.  The  damping 
ratio  in  Eq.  (A. 3)  is  now  assumed  to  be  the  same  (Ci)  for  all  modes,  since 
the  differences  in  the  ratio  are  usually  small  and  this  simplifies  the  equation. 
Taking  the  Fourier  transform  of  Eq.  (A.3)  with  zero  initial  conditions  leads 
to  the  following 

=  7i, +  S:$i,(ai)fETr2, $2,(02) 

t  r=0  i=0  -* 

</>!;•  =  K,  -  •  2(iu;i^iv){2  -  sgn(i)) 

J  fori  =  0 

where  (~)  denotes  a  Fourier  transform.  Similarly,  applying  the  previous 
procedure  to  Eq.  (A.2), 

r  OO  CO  -1 

M2(^2.1V2.  =  ^^$2.(02)  EW"li®l.(<»0-EW"2,$2,(o2)  (A.5) 

t  i=0  r=0 

^2s  =  {^2,  -  •  2C2^^2,w)(2  -  sgn(s)). 

Solving  for  VF2,  from  Eq.  (A.4)  and  (A.5), 

=  Trr-Wx-l^^-  -  (a.6) 


where 


_  W2.(a2).^/ii$i,(ai) 


1193 


Calculating  the  second  term  in  brackets  in  Eq.  (A. 6), 
fc4'2.(<22)EW^2,«'2.(a2)  E 


where 


Plugging  Eq.  (A. 7)  into  Eq.  (A. 6), 

M2<j>2s  1  +  CKl  +  Q;2  1=0 

\  I '  I.  -I—  ✓ 

Coupling  force  Ps 

Using  the  definition  of  transmitted  power  in  Ref.  [7],  1112(0))  is 


ni2  =  Re[-v^o)f;E[P.F;,(o))] 

L  „_ri 


"I"  0^1  +  P  i~0  j=:0  J 

A 

i<?^2.p  hh  Ml, 

where  Re[]  denotes  the  real  part  of  the  argument,  *  is  a  complex  conjugate, 
and 

Finally,  it  is  assumed  that  the  modal  driving  forces,  f-^.,  are  uncorrelated. 
Also,  the  spectral  density  function  of  each  modal  driving  force  is  assumed 
to  be  constant  for  the  finite  frequency  range  of  interest: 


C  (  \  f  ^PlVl  ^ 

V  for  i^j. 


Therefore,  Eq.  (A. 9)  becomes 


2^2fc^O)^.5^pipi  ^  ^  ^2r(^2)o)2r 

to  to 


(A.IO) 


(A.ll) 


1194 


PASSIVE  AND  ACTIVE  CONTROL  III 


Research  on  Control  Law  of  Active  Siispension  of  a 
Seven  Degree  of  Freedom  Vehicle  Model 

Dr&Prof.  YuchengLei  Lifen  Chen 
Automobile  Engineering  Dept,  Tong  Ji  University  ,Shang 
Hai,P.R.of  China 

Abstract 

In  the  paper  ,  control  law  of  active  suspension  is  presented  , 
which  involves  7-DOF  vehicle  model  for  improving  control 
accuracy  .The  control  law  involve  vehicle  running  velocity  , 
road  power  spectrum  ,  suspension  stiffness  and  damping  .The 
control  law  can  be  applied  to  multi-DOF  control  of  active 
suspension  of  vehicle  . 

Keywords:  Active  suspension  ,  control  law  ,  Game  theory  , 
Modeling  ,  7  —  DOF  Vehicle  Model . 

1 .  Introduction 

An  individual  control  system  for  each  wheel  by  applying  the 
optimum  regulation  method  for  the  two  degrees  of  freedom  is 
showed  in  [3]  .  [4]  and  [5]  also  introduce  two-DOF  feedback 
control  method  of  active  suspension  .It  is  difficult  for  two- 
DOF  control  method  to  coordinate  multi-DOF  kinematic 
distances  of  entire  car  .  Muti-DOF  active  control  can  improve 
coordination  control  accuracy  of  entire  car ,  but  high  speed  of 
CPU  is  asked  for  control  and  calculation  while  control  law  of 
multi-DOF  is  got  by  real-time  calculation  .  And,  ride 
performance  and  handling  performance  is  inconsistent  .  For 
resolving  the  problem  ,the  paper  holds  a  new  calculation 
method  for  optimizing  the  law  that  can  be  programmed  for 
real-time  control  by  table-lookup  and  not  by  real-time 
calculation  .So  the  method  and  law  can  not  only  improve 
coordination  control  accuracy  ,but  also  develop  control 
speed . 

2 .  Mathematical  Model 

Vehicle  is  simplified  to  turn  into  7  DOF  model  .  7-DOF 
vibration  motion  equation  can  be  written  as  follow 

[Af  ]z+  [c]z+  [kY  =  \C,  ](2+  ]e  (1) 


1195 


Where  [M]  is  mass  matrix  ,  [C]  is  suspension  damping 

matrix  ,  [A’]  is  suspension  stiffness  matrix  ,  [C,]  is  tyre 

•  • 

damping  matrix  ,  \K,  ]  is  tyre  stiffness  matrix  ,  Z  is 
acceleration  matrix  ,Z  is  velocity  matrix  ,  Z  is  7-DOF 


displacement  matrix ,  Q  is  road  surface  input  velocity  matrix , 
Q  is  road  surface  input  displacement  matrix . 

z=p„z....,zy  (2) 


••  .  •  -j-J 

Where  Zi  is  vehicle  vertical  acceleration,  Zi  is  roll 

••  •• 

acceleration  ,  Z3  is  pitch  acceleration  ,  Z,-  (/  =  4,*«*,7)  is 

four  tyres  vertical  acceleration  . 

3  General  Optimization  Method  of  Control  Law 
Objective  function  of  optimization  of  control  law  can 
generally  got  by  calculating  weighted  sum  of  7-DOF  mean 
square  root  of  acceleration  ,  dynamic  deflection  and  dynamic 
load  .it  can  be  written  as  follows 

/=!  >1  A=1 

Where  a-^  (z  =  !,•  •  (7  =  !,•  •  •A),  r  *  =  h*  •  •A)  is 

weighted  ratio  .  Where  a..  (i=l,  *  *  '  ,7)  is  7-DOF  mean 

Zi 

square  root  of  acceleration  ,  o-jy_  0=1?  ‘  '  is  7-DOF  mean 
square  root  of  dynamic  deflection  ,  (k=l,  *  *  ^  A)  is 

7-DOF  mean  square  root  of  relative  dynamic  load  .cr.. , 

Zi 

can  be  calculated  by  resolving  (1)  using  numerical 

method . 

4  Result  of  General  Optimization 
Method  of  Control  Law 

Optimization  result  of  control  law  of  a  truck  is  got  using 
above  method  as  figure  1  and  2  ,  its  main  parameters  as 
follows . 

Wheel  distance  is  1.4  meter  ,  axle  distance  is  2.297  meter  , 
mass  is  1121.3  kg  ,  front  tyre  and  axle’ s  mass  is  22.8  kg  , 
rear  tyre  and  axle’ s  mass  is  35.0  kg  ,  X  axis’  rotational 


1196 


inertia  is  307.4  kg-ni^  ,  Y  axis’  rotational  inertia  is  1276.5 
kg-rri^  . 

In  fig.  1  and  2  ,  Cl  of  RMSMIN  and  C3  of  RMSMIN  are 
respectively  front  and  rear  suspension  damping  of  getting 
minimization  of  above  objective  function  ,  it  is  changing 
while  road  surface  rough  coefficient  and  automobile 

velocity  V  is  changing  .  Cl  and  C3  also  rise  when  velocity  V 
rises  .  This  is  called  control  law  of  general  optimization 
method  of  active  suspension  in  the  paper  .  The  result  in  fig.  1 
and  fig.  2  has  been  verified  by  road  test . 


Fig.  1  front  suspension  Fig.  2  rear  suspension 

optimization  damping  optimization  damping 

Simulation  result  can  also  verify  that  ride  performance’s 
increasing  (suspension  stiffness  reducing)  will  make  handling 
performace  reduce  .  So  selecting  perfect 

=  is  very  difficult 

and  inconsistent  .The  paper  advances  next  game  method  to 
try  to  resolve  the  inconsistent  problem  . 

5  Game  Optimization  Method  of  Control  Law 
Because  to  select  weighted  ratio  of  general  optimization 
method  is  difficult  ,  the  paper  advances  a  new  method  of 
optimization  of  control  law  —  Game  Balance  Optimization 
Method  4t  is  discussed  as  follows  . 

Game  theory  method  of  two  countermeasure  aspect  can  be 
expressed  as  follows : 


1197 


—  1  j  ^  0  ,z  •—  1,  2, 

tr  (4) 

^  hj  —  I  ,  hj  ^0  ,  y  =  1,  2,-  •  *,  WZ2 

where  . r^,  is  probability  of  selecting  R^,R2> . of 

countermeasure  R  (  where  R  is  acceleration  mean  square 

root  )  ,  and  h,,h^, . X,  is  probability  of  selecting 

. ,  of  countermeasure  H  (  where  H  is 

mean  square  root  of  deflection  or  handling  and 

satiability  )  . 

It  is  called  hybrid  game  method  while  these  probability  is 
leaded  into  the  method .  Countermeasure  R  selects  in  order 
to  get  maximization  of  minimization  paying  expected  value 
of  column  vector  of  paying  matrix  ,  and  countermeasure  H 
selects  hj  in  order  to  get  minimization  of  maximization 
paying  expected  value  of  row  vector  of  paying  matrix  . 


If  rank  of  paying  matrix  is  x  ,R  should  select  r.,  as 
follows. 


max\min\ 


(5) 


relative 


And  H  should  select  hj  as  follows  : 


MAXl  MINI 


j=i 


(6) 


relative  hj 

a.j  (z  =  l, . ,y  =  l, . ,^2)  in  (5)  and  (6)  is  element 

value  of  paying  matrix  ,  basing  vehicle  theory  it  can  be  got 
as  follows  : 


a.j  =  C,  !  +  (7) 

Where  C^,C^  in  (7)  is  coefficient  of  paying  matrix  (  The 
paper  orders  they  is  1  as  an  example  ,  as  Q  ,C2  ’  s  real  value 
about  very  much  condition  is  related  to  some  privacy 


1198 


problem  it  can’ t  be  introduced. )  .  (5), (6)  called  respectively 
minimization  maximization  expected  value  and  maximization 
minimization  expected  value  can  be  abbreviated  as 
MAxi^IN^^  )}  and  MIN  ^AX(^  )}  .  if  r.,andhj  is 

got  as  optimization  of  countermeasure  ,  it  can  be  wntten  as 
follows  : 

MAX^IN^  ^^Optimization  Countermeasure 

Expected  Value  ^  MIN 

A  probability  association  {r^ Xj)  can  be  content  with 

optimization  expected  value  as  follows  : 

Optimization  Countermeasure  Expected 

m^  m-i 

Value=|]  J^aij»r;»hj  (9) 

/=! 

Writer  advances  reformation  simplex  algorithm  for  resolving 
the  game  problem  as  reference  [1]  .  In  the  paper  the  water 
selects  only  an  example  to  introduce  calculation  results  as 
follows  because  the  paper  has  limited  space  . 

6  Results  of  Game  Optimization  Method  of  Control  Law 
Paying  matrix  as  fig.  3  and  fig.  4  ,  optimization  result  of 
control  law  of  a  truck  is  got  using  above  game  method  as 
figure  5  and  6  ,  calculated  truck’s  main  parameters  as 
follows . 

Wheel  distance  is  1.23  meter  ,  axle  distance  is  3.6  meter  , 
mass  is  13880.0  kg  ,  front  tyre  and  axle’ s  mass  is  280.0  kg  , 
rear  tyre  and  axle’ s  mass  is  280.0  kg  ,  X  axis’  rotational 
inertia  is  1935  kg^m^  ,  Y  axis’  rotational  inertia  is  710 
kg-m-  . 

In  fig.  3  and  4  3(IJ)  is  paying  matrix  value 

{i  =  1, . ,m,  J  =  1, . ,m,) .  In  fig.  5  and  fig.  6  ,  K1  of 

RMSMIN  and  K3  of  RMSMIN  are  respectively  front  and 
rear  suspension  stiffness  of  getting  optimization 
countermeasure  expected  value  of  above  game  method  ,  it  is 
changing  while  road  surface  rough  coefficient  Q  and 

automobile  velocity  V  is  changing  .  K1  and  K3  also  rise 


1199 


when  velocity  V  rises  .  This  is  called  control  law  of  active 
suspension  .  The  result  in  fig.  5  and  fig.  6  has  been  verified 
by  road  test . 


Fig.  3  paying  matrix  Fig.  4  paying  matrix 

7  Conclusion 

The  paper  introduces  two  method  to  get  optimization  control 


Fig.  5  front  suspension  Fig.  6  rear  suspension 

optimization  stiffness  optimization  stiffness 

law  of  active  suspension  ,  and  the  control  law  is  verified  to 
ability  to  be  applied  to  real  control  of  active  suspension  .  This 
will  develop  control  accuracy  and  speed  of  active 
suspension  .Off  course  ,  it  need  being  researched  further  . 

8  reference 

1  Lei  Yucheng  ,  Theory  and  Engineering  Realization  of 
Semi  —  Active  Control  of  Vehicle  Vibration  ,Dr.  paper  , 
Harbin  Institute  of  Technology , China  ,1995  6  . 

2  Thompson  A.G.  ,  A  Suspension  Proc.  Int  of  Mechanical 


1200 


Engr.  Vol  185  No. 36, 970  —  990,553  563  . 

3  Lei ,  S. ,  Fasuda  ,  E.  and  Hayashi ,  Y.  :  “An  Experimental 
Study  of  Optimal  Vibration  Adjustment  Using  Adaptive 
Control  Methods  ”,  Proc  .  IMechE  Int  .  Conf  .  Advanced 
Suspensions  ,  London  ,  England  ,  (1988)  ,  C433/88  ,  119- 
124. 

4  Kamopp  D.  , Active  Damping  in  Road  Vehicle  Systems  , 
VSD,  12(1983),  291-316. 

5  Kamopp  D  .  C  .,  Grosby  M.  J.  &  Harword  R.  ,Vibration 
Control  Using  Semi-Active  Force  ,  Generator  ,  Trans  . 
ASME,  J  .  Eng  .  for  Ind  .  Vol .  96  Ser  .  B  ,  No  .2  ,  (1974) , 
619-626 . 


1201 


1202 


Designing  Heavy  Truck  Suspensions  for  Reduced  Road  Damage 

Mehdi  Ahmadian 
Edward  C.  Mosch  Jr. 


Department  of  Mechanical  Engineering 
Virginia  Polytechnic  Institute  and  State  University 
Blacksburg,  Virginia  24061-0238;  USA 
(540)  231-4920/-9100(fax) 
ahmadian@vt.edu 


ABSTRACT 

The  role  of  semiactive  dampers  in  reducing  tire  dynamic  loading  is  examined. 
An  alternative  to  the  well-known  skyhook  control  policy,  called 
“groundhook,”  is  introduced.  Using  the  dynamic  model  of  a  single  suspension, 
it  is  shown  that  groundhook  semiactive  dampers  can  reduce  tire  dynamic 
loading,  and  potentially  lessen  road  damage,  for  heavy  trucks. 


INTRODUCTION 

The  main  intent  of  this  work  is  to  determine,  analytically,  the  role  of 
semiactive  suspension  systems  in  reducing  tire  dynamic  loading,  and  road  and 
bridge  damage.  Although  primary  suspension  systems  with  semiactive 
dampers  have  been  implemented  in  some  vehicles  for  improving  ride  and 
handling,  their  impact  on  other  aspects  of  the  vehicle  remain  relatively 
unknown.  Specifically,  it  is  not  yet  known  if  implementing  semiactive 
dampers  in  heavy  truck  suspension  systems  can  reduce  the  tire  dynamic 
forces  that  are  transferred  by  the  vehicle  to  the  road.  Reducing  dynamic  forces 
will  result  in  reducing  pavement  loading,  and  possibly  road  and  bridge  damage. 

The  idea  of  semiactive  dampers  has  been  in  existence  for  more  than 
two  decades.  Introduced  by  Karnopp  and  Crosby  in  the  early  70’ s  [1-2], 
semiactive  dampers  have  most  often  been  studied  and  used  for  vehicle  primary 
suspension  systems.  A  semiactive  damper  draws  small  amounts  of  energy  to 
operate  a  valve  to  adjust  the  damping  level  and  reduce  the  amount  of  energy 
that  is  transmitted  from  the  source  of  vibration  energy  (e.g.,  the  axle)  to  the 
suspended  body  (e.g.,  the  vehicle  structure).  Therefore,  the  force  generated  by 
a  semiactive  damper  is  directly  proportional  to  the  relative  velocity  across  the 
damper  (just  like  a  passive  damper).  Another  class  of  dampers  that  is  usually 
considered  for  vibration  control  is  fully  active  dampers.  Active  dampers  draw 

1203 


relatively  substantial  amounts  of  energy  to  produce  forces  that  are  not 
necessarily  in  direct  relationship  to  the  relative  velocity  across  the  damper. 

The  virtues  of  active  and  semiactive  dampers  versus  passive  dampers 
have  been  addressed  in  many  studies  [3-10].  Using  various  analytical  and 
experimental  methods,  these  studies  have  concluded  that  in  nearly  all  cases 
semiactive  dampers  reduce  vibration  transmission  across  the  damper  and 
better  control  the  suspended  (or  sprung)  body,  in  comparison  to  passive 
dampers.  Further,  they  have  shown  that,  for  vehicle  primary  suspension 
systems,  semiactive  dampers  can  lower  the  vibration  transmission  nearly  as 
much  as  fully  active  dampers;  without  the  inherent  cost  and  complications 
associated  with  active  dampers.  This  has  led  to  the  prototype  application, 
and  production,  of  semiactive  dampers  for  primary  suspensions  of  a  wide 
variety  of  vehicles,  ranging  from  motorcycles,  to  passenger  cars,  to  bus  and 
trucks,  and  to  military  tanks,  in  favor  of  fully  active  systems. 

Although  there  is  abundant  research  on  the  utility  of  semiactive 
dampers  for  improving  vehicle  ride  and  handling,  their  potential  for  reducing 
dynamic  forces  transmitted  to  the  pavement  remains  relatively  unexplored. 
This  is  because  most  suspension  designers  and  researchers  are  mainly 
concerned  with  the  role  of  suspension  systems  from  the  vehicle  design 
perspective.  Another  perspective,  however,  is  the  effect  of  suspension 
systems  on  transmitting  dynamic  loads  to  the  pavement. 


ROAD  DAMAGE  STUDIES 

Dynamic  tire  forces,  that  are  heavily  influenced  by  the  suspension,  are 
believed  to  be  an  important  cause  of  road  damage.  Cole  and  Cebon  [11] 
studied  the  design  of  a  passive  suspension  that  causes  minimum  road  damage 
by  reducing  the  tire  force.  They  propose  that  there  is  a  stronger  correlation 
between  the  forth  power  of  the  tire  force  and  road  damage  than  the  dynamic 
load  coefficient  (DLC)  and  road  damage. 

A  simple  measure  of  road  damage,  introduced  by  Cebon  m  [12],  is  the 
aggregate  fourth  power  force  defined  as 

Na 

Al=Y,Pjk  k=  1,2,3  ..ns  (1) 

where  Pjk  =  force  applied  by  tire  j  to  point  k  along  the  wheel  path, 
ria  -  number  of  axles  on  vehicle,  and 

Us  =  number  of  points  along  the  road.. 


1204 


DLC  is  a  popular  measurement  frequently  used  to  characterize  dynamic 
loading  and  is  defined  as  the  root  mean  square  (RMS)  of  the  tire  force  divided 
by  the  mean  tire  force,  which  is  typically  the  static  weight  of  the  vehicle.  The 
equation  takes  the  form; 


RMS  Dynamic  Tire  Force 
Static  Tire  Force 

This  study  shows  that  minimum  road  damage,  for  a  two  degree-of-freedom 
model,  is  achieved  by  a  passive  system  with  a  stiffness  of  about  one  fifth  of 
current  air  suspensions  and  a  damping  of  about  twice  that  typically  provided. 
In  practice,  however,  reducing  the  suspension  stiffiiess  can  severely  limit  the 
static  load  carrying  capacity  of  the  suspension  and  cause  difficulties  in  vehicle 
operation.  Further,  higher  damping  can  substantially  increase  vibration 
transmission  to  the  body  and  worsen  the  ride. 

In  another  study  by  Cole  and  Cebon  [13]  a  two-dimensional  articulated 
vehicle  simulation  is  validated  with  measurements  from  a  test  vehicle.  The 
effect  of  modifications  to  a  trailer  suspension  on  dynamic  tire  forces  are 
investigated.  The  RMS  of  dynamic  loads  generated  by  the  trailer  are  predicted 
to  decrease  by  3 1  per  cent,  resulting  in  a  predicted  decrease  in  theoretical  road 
damage  of  about  13  per  cent. 

Yi  and  Hedrick  compared  the  effect  of  continuous  semiactive  and 
active  suspensions  and  their  effect  on  road  damage  using  the  vehicle  simulation 
software  VESYM  [14].  A  control  strategy  based  on  the  tire  forces  in  a  heavy 
truck  model  is  used  to  show  that  active  and  semiactive  control  can  potentially 
reduce  pavement  loading.  They,  however,  mention  that  measuring  the  tire 
forces  poses  serious  limitation  in  practice. 

The  primary  purpose  of  this  paper  is  to  extend  past  studies  on 
semiactive  suspension  systems  for  reducing  road  damage.  An  alternative 
semiactive  control  policy,  called  "groundhook,”  is  developed  such  that  it  can 
be  easily  applied  in  practice,  using  existing  hardware  for  semiactive 
suspensions.  A  simulation  model  representing  a  single  primary  suspension  is 
used  to  illustrate  the  system  effectiveness.  The  simulation  results  show  that 
groundhook  control  can  reduce  the  dynamic  load  coefficient  and  fourth  power 
of  tire  force  substantially,  without  any  substantial  increase  in  body 
acceleration. 


1205 


MATHEMATICAL  FORMULATION 


We  consider  a  model  representing  the  dynamics  of  a  single  primary  suspension 
in  a  heavy  truck,  as  shown  in  Figure  1. 


Truck  Mass 


Suspension 

Stiffness 

Suspension 
and  Tire  Mass 


Tire 

Stiffness 


Suspension 

Damping 


Road  Input 


Figure  1.  Primary  Suspension  Model 

This  model  has  been  widely  used  in  the  past  for  automobile  applications,  due 
to  its  effectiveness  in  analyzing  various  issues  relating  to  suspensions. 
Although  it  does  not  include  the  interaction  between  the  axles  and  the  truck 
frame  dynamics,  the  model  still  can  serve  as  an  effective  first  step  in  studying 
fundamental  issues  relating  to  truck  suspensions.  Follow  up  modeling  and 
testing,  using  a  full  vehicle,  is  needed  to  make  a  more  accurate  assessment. 


The  dynamic  equations  for  the  model  in  Figure  1  are: 


M^x^+C(x^-X2)  + K(x^~X2)  =  0  (3a) 

MjX,  - C(ii  -X2)-K{x^  - ^2)  +  0 

The  variables  Xi  and  X2  represent  the  body  and  axle  vertical  displacement, 
respectively.  The  variable  Xjn  indicates  road  input,  that  is  assumed  to  be  a 
random  input  with  a  low-pass  (0  -  25  Hz)  filter.  The  amplitude  for  Xjn  is 
adjusted  such  that  it  creates  vehicle  and  suspension  dynamics  that  resembles 
field  measurements.  Such  a  function  has  proven  to  sufficiently  represent 
actual  road  input  to  the  vehicle  tires. 

Table  1  includes  the  model  parameters,  that  are  selected  to  represent  a 
typical  laden  truck  used  in  the  U.S.  The  suspension  is  assumed  to  have  a 
linear  stiffness  in  its  operating  range.  The  damper  characteristics  are  modeled 
as  a  non-linear  function,  as  shown  in  Figure  2. 


1206 


Table  1.  Model  Parameters 


Description 

Symbol 

Value 

Body  Mass 

Ml 

287  kg 

Axle  Mass 

M2 

34  kg 

Suspension  Stiffness 

K1 

196,142  N/m 

Tire  Stifftiess 

Kt 

1,304,694  N/m 

Suspension  Damping 

C 

See  Table  2 

The  bilinear  function  in  Figure  2  represents  the  force-velocity 
characteristics  of  an  actual  truck  damper.  The  parameters  selected  for  both 
passive  and  semiactive  dampers  are  shown  in  Table  2.  These  parameters  are 
selected  based  on  truck  dampers  commonly  used  in  the  U.S.  Although  we 
examined  the  effect  of  damper  tuning  on  dynamic  loading,  if  falls  outside  the 
scope  of  this  paper.  Instead,  we  concentrate  here  on  comparing  different 
semiactive  dampers  with  a  passive  damper,  using  the  baseline  parameters 
shown  in  Table  2.  The  semiactive  and  passive  damper  characteristics  used  for 
this  study  are  further  shown  in  Figure  3. 


1207 


Force  Velocity  Curve 


Figure  3.  Passive  and  Semiactive  Damper  Characteristics 


SKYHOOK  CONTROL  POLICY 

As  mentioned  earlier,  the  development  of  semiactive  dampers  dates  back  to 
early  70’s  when  Kamopp  and  Crosby  introduced  the  skyhook  control  policy. 
For  the  system  shown  in  Figure  1,  skyhook  control  implies 

X,(X]  -  ;C2 )  -  0  C  =  Con 
X](X^-X2)<0  C  =  Coff 

Where  x,  and  represents  the  velocities  of  Mi  (vehicle  body)  and  M2  (axle), 
respectively.  The  parameters  Con  and  Coff  represent  the  on-  and  off-state  of 
the  damper,  respectively,  as  it  is  assumed  that  the  damper  has  two  damping 
levels.  In  practice,  this  is  achieved  by  equipping  the  hydraulic  damper  with  an 
orifice  that  can  be  driven  by  a  solenoid.  Closing  the  orifice  increases  damping 
level  and  achieves  Con,  whereas  opening  it  gives  Coff . 


1208 


Table  2.  Damper  Parameters 


Passive 

Semiactive  On-State 

Semiactive  Off-State 

n 

0.25 

0.35 

0.03 

m 

0.10 

0.15 

0.03 

Vbb 

0.254  m/sec 

0.254  m/sec 

0.254  m/sec 

B 

0.20 

0.30 

0.03 

S4 

0.10 

0.15 

0.03 

Vbj 

0.254  m/sec 

0.254  m/sec 

0.254  m/sec 

The  switching  between  the  two  damper  states,  shown  in  Eq.  (4),  is 
arranged  such  that  when  the  damper  is  opposing  the  motion  of  the  sprung 
mass  (vehicle  body),  it  is  on  the  on-state.  This  will  dampen  the  vehicle  body 
motion.  When  the  damper  is  pushing  into  the  body,  it  is  switched  to  the  off- 
state  to  lower  the  amount  of  force  it  adds  to  the  body.  Therefore,  a  semiactive 
damper  combines  the  performance  of  a  stiff  damper  at  the  resonance 
frequency,  and  a  soft  damper  at  the  higher  frequencies,  as  shovm  in  Figure  4. 


1209 


Figure  5.  Groundhook  Damper  Configurations:  a)  optimal  groundhook 
damper  configuration,  b)  semiactive  groundhook  damper 
configuration. 

This  feature  allows  for  a  better  control  of  the  vehicle  body,  as  has  been 
discussed  in  numerous  past  studies.  The  skyhook  control  policy  in  Eq.  (4), 
however,  works  such  that  it  increases  axle  displacement,  X2,  (commonly  called 
wheel  hop).  Because  the  tire  dynamic  loading  can  be  defined  as 


DL  =  KtX2  (5) 

The  skyhook  control  actually  increases  dynamic  loading.  As  mentioned  earlier 
the  development  of  skyhook  policy  was  for  improving  ride  comfort  of  the 
vehicle,  without  losing  vehicle  handling.  Therefore,  the  dynamic  loading  of  the 
tires  was  not  a  factor  in  the  control  development. 


GROUNDHOOK  CONTROL  POLICY 

To  apply  semiactive  dampers  to  reducing  tire  dynamic  loading,  we  propose  an 
alternative  control  policy  that  can  be  implemented  in  practice  using  the  same 
hardware  needed  for  the  skyhook  policy.  To  control  the  wheel  hop,  this 
policy,  called  “groundhook,”  implies: 


X,  (;ci  -  X2 )  <  0  C  =  Con 

(6a) 

x,  (Xt  -  ^2 )  >  0  C  =  Coff 

(6b) 

As  shovm  in  Figure  5,  the  above  attempts  to  optimize  the  damping  force  on 
the  axle,  similar  to  placing  a  damper  between  the  axle  and  a  fictitious  ground 
(thus,  the  name  “groundhook”).  The  groundhook  semiactive  damper 
maximizes  the  damping  level  (i.e.,  C  =  Con)  when  the  damper  force  is  opposing 


1210 


the  motion  of  the  axle;  otherwise,  it  minimizes  the  damping  level  (i.e.,  C  = 
Coff).  The  damper  hardware  needed  to  implement  groimdhook  semiactive  is 
exactly  the  same  as 

the  skyhook  semiactive,  except  for  the  control  policy  programmed  into  the 
controller. 


SIMULATION  RESULTS 

The  model  shown  in  Figure  1  is  used  to  evaluate  the  benefits  of  groundhook 
dampers  versus  passive  and  skyhook  dampers.  A  non-linear  damper  model 
was  considered  for  the  simulations,  as  discussed  earlier.  The  road  input  was 
adjusted  such  that  the  dynamic  parameters  for  the  passive  damper  resembles 
actual  field  measurements.  Five  different  measures  were  selected  for 
comparing  the  dampers: 

•  Dynamic  Load  Coefficient  (DLC) 

•  Fourth  Power  of  the  tire  dynamic  load 

•  Sprung  mass  acceleration 

•  Rattle  Space  (relative  displacement  across  the  suspension) 

•  Axle  Displacement,  relative  to  the  road 

Dynamic  load  coefficient  and  fourth  power  of  tire  force  are  measures 
of  pavement  dynamic  loading  and  are  commonly  used  for  assessing  road 
damage.  They  are  both  considered  here  because  there  is  no  clear  consensus  on 
which  one  is  a  better  estimate  of  road  damage.  Axle  displacement,  relative  to 
the  road,  indicates  wheel  hop  and  is  directly  related  to  DLC  and  tire  force, 
therefore  it  is  yet  another  measure  of  road  damage.  Sprung  mass  acceleration 
is  a  measure  of  ride  comfort.  Our  experience,  however,  has  shown  that  for 
trucks  this  may  not  be  a  reliable  measure  of  the  vibrations  the  driver  feels  in 
the  truck.  The  relative  displacement  across  the  dampers  relates  to  the  rattle 
space,  that  is  an  important  design  parameter  in  suspension  systems, 
particularly  for  cars.  For  each  of  the  above  measures,  the  data  was  evaluated 
in  both  time  (Figures  6-7)  and  frequency  domain  (Figures  8  -  10).  In  time 
domain,  the  root  mean  square  (RMS)  and  maximum  of  the  data  for  a  five- 
second  simulation  are  compared.  In  frequency  domain,  the  transfer  function 
between  each  of  the  measures  and  road  displacement  is  plotted  vs.  frequency. 
The  frequency  plots  highlight  the  effect  of  each  damper  on  the  body  and  axle 
resonance  frequency. 

Figures  6  and  7  show  bar  charts  of  root  mean  square  (RMS)  and 
maximum  time  data,  respectively.  In  each  case  the  data  is  normalized  with 
respect  to  the  performance  of  passive  dampers  commonly  used  in  trucks. 
Therefore,  values  below  line  1 .0  can  be  interpreted  as  an  improvement  over  the 
existing  dampers.  As  Figures  6  and  7  show,  groundhook  dampers  significantly 


1211 


improve  pavement  loading,  particularly  as  related  to  the  fourth  power  of  tire 
force.  Furthermore,  the  rattle  space  is  improved  slightly  over  passive 
dampers,  indicating  that  groundhook  dampers  do  not  impose  any  additional 
burden  on  the  suspension  designers. 

One  measure  that  has  increased  due  to  groundhook  dampers  is  body 
acceleration.  As  mentioned  earlier,  in  automobiles  this  measure  is  used  as  an 
indicator  of  ride  comfort.  In  our  past  testing,  however,  we  have  found  that  for 
trucks  it  is  a  far  less  accurate  measure  of  ride  comfort.  This  is  mainly  due  to 
the  complex  dynamics  of  the  truck  frame  and  the  truck  secondary  suspension. 
A  more  accurate  measure  of  ride  comfort  is  acceleration  at  the  B-Post  (the 
post 


Tire  Axle  Body  Rattle  Space 

Dynamic  Displacment  Acceleration 

Force  ^  4 


Figure  6.  RMS  Time  Data  Normalized  with  respect  to  Passive  Damper 


Tire  Axle  Body  Rattle  Space 

Dynamic  Displacment  Acceleration 

Force  4 


Figure  7.  Max.  Time  Data  Normalized  with  respect  to  Passive  Damper 

1212 


behind  the  driver),  which  cannot  be  evaluated  from  the  single  suspension 
model  considered  here.  Nonetheless,  the  body  acceleration  is  included  for  the 
sake  of  completeness  of  data. 

The  model  shows  that  skyhook  dampers  actually  increase  the 
measures  associated  with  pavement  loading,  while  improving  body 
acceleration.  This  agrees  with  the  purpose  of  skyhook  dampers  that  are 
designed  solely  for  improving  the  compromise  between  ride  comfort  and 
vehicle  handling.  The  improvement  in  ride  comfort  occurs  at  the  expense  of 
increased  pavement  loading. 


Figure  8.  Transfer  Function  between  Axle  Displacement  and  Road  Input 


Figures  8-10  show  the  frequency  response  of  the  system  due  to  each  damper. 
In  each  figure,  the  transfer  fimction  between  one  of  the  measures  and  input 
displacement  is  plotted  vs.  frequency.  These  plots  highlight  the  impact  of 
skyhook  and  groundhook  on  the  body  and  wheel  hop  resonance,  relative  to 
existing  passive  dampers.  The  frequency  plots  indicate  that  the 


1213 


Transfer 


Frequency  (Hz) 


groundhook  dampers  reduce  axle  displacement  and  fourth  power  of  tire 
dynamic  force  at  wheel  hop  frequency.  At  body  resonance  frequency, 
groundhook  dampers  do  not  offer  any  benefits  over  passive  dampers.  The 


frequency  results  for  body  acceleration  and  rattle  space  are  similar  to  those 
discussed  earlier  for  the  time  domain  results.  The  frequency  plots  show  that 
the  skyhook  dampers  offer  benefits  over  passive  dampers  at  frequencies  close 
to  the  body  resonance  frequencies.  At  the  higher  frequencies,  associated  with 
wheel  hop,  skyhook  dampers  result  in  a  larger  peak  than  either  passive  or 
groundhook  dampers.  This  indicates  that  skyhook  dampers  are  not  suitable 
for  reducing  tire  dynamic  loading. 


CONCLUSIONS 

An  alternative  to  skyhook  control  policy  for  semiactive  dampers  was 
developed.  This  policy,  called  “groundhook,”  significantly  improves  both 
dynamic  load  coefficient  (DLC),  and  fourth  power  of  tire  dynamic  load, 
therefore  holding  a  great  promise  for  reducing  road  damage  to  heavy  trucks. 
The  dynamic  model  used  for  assessing  the  benefits  of  groundhook  dampers 
represented  a  single  suspension  system.  Although  the  results  presented  here 
show  groundhook  dampers  can  be  effective  in  reducing  tire  dynamic  loading 
and  pavement  damage,  more  complete  models  and  road  testing  are  necessary 
for  more  accurately  assessing  the  benefits. 


REFERENCES 

1.  Crosby,  M.  J.,  and  Karnopp,  D.  C.,  "The  Active  Damper,"  The  Shock 
and  Vibration  Bulletin  43,  Naval  Research  Laboratory,  Washington,  D.C., 
1973. 

2.  Karnopp,  D.  C.,  and  Crosby,  M.  J.,  "System  for  Controlling  the 
Transmission  of  Energy  Between  Spaced  Members,"  U.S.  Patent 
3,807,678,  April  1974. 

3.  Ahmadian,  M.  and  Marjoram,  R.  H.,  “Effects  of  Passive  and  Semi-active 
Suspensions  on  Body  and  Wheelhop  Control,”  Journal  of  Commercial 
Vehicles,  Vol.  98,  1989,  pp.  596-604. 

4.  Ahmadian,  M.  and  Marjoram,  R.  H.,  “On  the  Development  of  a 
Simulation  Model  for  Tractor  Semitrailer  Systems  with  Semiactive 
Suspensions,”  Proceedings  of  the  Special  Joint  Symposium  on  Advanced 
Technologies,  1989  ASME  Winter  Annual  Meeting,  San  Francisco, 
California,  December  1989  (DSC-Vol.  13). 

5.  Hedrick,  J.  K.,  "Some  Optimal  Control  Techniques  Applicable  to 
Suspension  System  Design,"  American  Society  of  Mechanical  Engineers, 
Publication  No.  73-ICT-55,  1973. 


1215 


6.  Hac,  A.,  "Suspension  Optimization  of  a  2-DOF  Vehicle  Model  Using 
Stochastic  Optimal  Control  Technique,"  Journal  of  Sound  and  Vibration, 
1985. 

7.  Thompson,  A.  G.,  "Optimal  and  Suboptimal  Linear  Active  Suspensions 
for  Road  Vehicles,"  Vehicle  System  Dynamics,  Vol.  13, 1984. 

8.  Kamopp,  D.,  Crosby,  M.  J.,  and  Harwood,  R.  A.,  "Vibration  Control 
Using  Semiactive  Force  Generators,"  American  Society  of  Mechanical 
Engineers,  Journal  of  Engineering  for  Industry,  May  1974,  pp.  619-626. 

9.  Krasnicki,  E.  J.,  "Comparison  of  Analytical  and  Experimental  Results  for 
a  Semiactive  Vibration  Isolator,"  Shock  and  Vibration  Bulletin,  Vol.  50, 
September  1980. 

10.  Chalasani,  R.M.,  "Ride  Performance  Potential  of  Active  Suspension 
Systems-Part  1:  Simplified  Analysis  Based  on  a  Quarter-Car  Model," 
proceedings  of  1986  ASME  Winter  Annual  Meeting,  Los  Angeles,  CA, 
December  1986. 

11.  Cole,  D.  J.  and  Cebon,  D.,  “Truck  Suspension  Design  to  Minimize  Road 
Damage,”  Proceedings  of  the  Institution  of  Mechanical  Engineers,  Vol. 
210,  D06894,  1996,  pp.  95-107. 

12.  Cebon,  D.,  “Assessment  of  the  Dynamic  Forces  Generated  by  Heavy 
Road  Vehicles,”  ARRB/FORS  Symposium  on  Heavy  Vehicle  Suspension 
Characteristics,  Canberra,  Australia,  1987. 

13.  Cole,  D.  J.  and  Cebon,  D.,  “Modification  of  a  Heavy  Vehicle  Suspension 
to  Reduce  Road  Damage,”  Proceedings  of  the  Institution  of  Mechanical 
Engineers,  Vol.  209,  D03594,  1995. 

14.  Yi,  K.  and  Hedrick,  J.  K.,  “Active  and  Semi- Active  Heavy  Truck 
Suspensions  to  Reduce  Pavement  Damage,”  SAE  SP-802,  paper  892486, 
1989. 


1216 


Active  Vibration  Control  of  Isotropic  Plates  Using 
Piezoelectric  Actuators 


A.  M.  Sadri",  J.  R.,  Wright*  and  A.  S.  Cherry* 

The  Manchester  School  of  Engineering,  Manchester  M13  9PL,  UK 

and 

R.  J.  Wynnes 

Sheffield  Hallam  University,  School  of  Engineering,  Sheffield,  UK 


Abstract:  Theoretical  modelling  of  the  vibration  of  plate  components  of  a 
space  structure  excited  by  piezoelectric  actuators  is  presented.  The  equations 
governing  the  dynamics  of  the  plate,  relating  the  strains  in  the  piezoelectric 
elements  to  the  strain  induced  in  the  system,  are  derived  for  isotropic  plates 
using  the  Rayleigh-Ritz  method.  The  developed  model  was  used  for  a  simply 
supported  plate.  The  results  show  that  the  model  can  predict  natural 
frequencies  and  mode  shapes  of  the  plate  very  accurately.  The  open  loop 
frequency  response  of  the  plate  when  excited  by  the  patch  of  piezoelectric 
material  was  also  obtained.  This  model  was  used  to  predict  the  closed  loop 
frequency  response  of  the  plate  for  active  vibration  control  studies  with 
suitable  location  of  sensor-actuators. 


Introduction 

Vibration  suppression  of  space  structures  is  very  important  because 
they  are  lightly  damped  due  to  the  material  used  and  the  absence  of  air 
damping.  Thus  the  modes  of  the  structure  must  be  known  very 
accurately  in  order  to  be  affected  by  the  controller  while  avoiding 
spillover.  This  problem  increases  the  difficulty  of  predicting  the 
behaviour  of  the  structure  and  consequently  it  might  cause  unexpected 
on-orbit  behaviour. 

These  difficulties  have  motivated  researchers  to  use  the 
actuation  strain  concept.  One  of  the  mechanisms  included  in  the 
actuation  strain  concept  is  the  piezoelectric  effect  whereby  the  strain 
induced  through  a  piezoelectric  actuator  is  used  to  control  the 


Research  Student,  Dynamics  &:  Control  Research  Group. 
^  Professor ,  Dynamics  &  Control  Research  Group, 

^  Former  Lecturer,  Dynamics  &  Control  Research  Group. 
^  Professor  of  Mechanical  and  Control  Engineering. 


1217 


deformation  of  the  structure  [1].  It  can  be  envisaged  that  using  this 
concept  in  conjimction  with  control  algorithms  can  enhance  the  ability 
to  suppress  modes  of  vibration  of  flexible  structures. 

Theoretical  and  experimental  results  of  the  control  of  a  flexible 
ribbed  antenna  using  piezoelectric  materials  has  been  investigated  in 
[2].  An  active  vibration  damper  for  a  cantilever  beam  using  a 
piezoelectric  polymer  has  been  designed  in  [3].  In  this  study, 
Lyapunov's  second  or  direct  method  for  distributed-parameter  systems 
was  used  to  design  control  algorithms  and  the  ability  of  the  algorithms 
was  verified  experimentally.  These  works  have  clearly  shown  the 
ability  of  piezoelectric  actuators  for  vibration  suppression.  However, 
they  have  been  limited  to  one  dimensional  systems.  Obviously,  there  is 
a  need  to  understand  the  behaviour  of  piezoelectric  actuators  in  two 
dimensional  systems  such  as  plates. 

Vibration  excitation  of  a  thin  plate  by  patches  of  piezoelectric 
material  has  been  investigated  in  [4].  Their  work  was  basically  an 
extension  of  the  one  dimensional  theory  derived  in  [1]  to  show  the 
potential  of  piezoelectric  actuators  in  two  dimensions.  In  their  studies, 
it  was  assumed  that  the  piezoelectric  actuator  doesn't  significantly 
change  the  inertia,  mass  or  effective  stiffness  of  the  plate.  This 
assumption  is  not  guaranteed  due  to  the  size,  weight  and  stiffness  of 
the  actuator.  Based  on  this  assumption,  their  model  can  not  predict  the 
natural  frequencies  of  the  plate  accurately  after  bonding  piezoelectric 
actuators.  Therefore,  it  is  essential  to  have  a  more  general  model  of  a 
plate  and  bonded  piezoelectric  actuators  with  various  boundary 
conditions.  The  model  should  be  able  to  predict  frequency  responses 
because  this  is  fundamental  to  the  understanding  of  the  behaviour  of 
the  system  for  control  design  purposes.  It  is  the  objective  of  the  current 
study  to  develop  such  a  modelling  capability. 

Previous  work  [5,  6,  7]  has  concentrated  on  the  modelling  and 
control  of  a  cantilever  beam.  The  method  used  involved  bonding 
piezoelectric  material  to  a  stiff  constraining  layer,  which  was  bonded  to 
the  beam  by  a  thin  viscoelastic  layer  in  order  to  obtain  both  active  and 
passive  damping.  Then  a  Rayleigh-Ritz  model  was  developed  and 
used  to  derive  a  linearized  control  model  so  as  to  study  different 
control  strategies.  In  the  work  described  in  this  paper,  the  method  has 
been  extended  to  the  more  complex  plate  problem.  The  paper 
introduces  a  modelling  approach  based  on  the  Rayleigh-Ritz  assumed 
mode  shape  method  to  predict  the  behaviour  of  a  thin  plate  excited  by 
a  patch  of  piezoelectric  material  bonded  to  the  surface  of  the  plate.  The 
model  includes  the  added  inertia  and  stiffness  of  the  actuator  and  has 
been  used  to  predict  the  frequency  response  of  the  plate.  Suggestions 
for  future  work  are  also  included. 


1218 


Theoretical  Modelling 

In  developing  the  Rayleigh-Ritz  model  of  a  plate  excited  by  a 
patch  of  piezoelectric  material  bonded  to  the  surface  of  the  plate,  a 
number  of  assumptions  must  be  made.  The  patch  of  piezoelectric 
material  is  assumed  to  be  perfectly  bonded  to  the  surface  of  the  plate. 
The  magnitude  of  the  strains  induced  by  the  piezoelectric  element  is  a 
linear  fimction  of  the  applied  voltage  that  can  be  expressed  by 


e:=E: 


(1) 


Here  is  the  piezoelectric  strain  constant,  is  the  piezoelectric  layer 

thickness  and  V33  is  the  applied  voltage.  The  index  31  shows  that  the 
induced  strain  in  the  '1'  direction  is  perpendicular  to  the  direction  of 
poling  '3'  and  hence  the  applied  field.  The  piezoelectric  element 
thickness  is  assumed  to  be  small  compared  to  the  plate  thickness.  The 
displacements  of  the  plate  middle  surface  are  assumed  to  be  normal  to 
it  due  to  the  bending  affects. 

Figure  1  shows  the  configuration  of  the  bonded  piezoelectric 
material  relative  to  the  surface  of  the  plate. 


w 


Figure  1.  Configuration  of  the  bonded  piezoelectric  actuator  on  the  surface  of  the 
plate. 

In  figure  1,  4  and  4  are  the  dimensions  of  the  plate,  x, ,  X2  ,  y,  and  y^ 
are  the  boundaries  of  the  piezoelectric  element  and  w,  v  and  w  are  the 
displacements  in  the  x  ,  y  and  z  direction,  respectively. 

To  derive  the  equations  of  motion  of  the  plate  based  on  the 
Rayleigh-Ritz  method,  both  the  strain  energy  U  and  kinetic  energy  T  of 
the  plate  and  the  piezoelectric  element  must  be  determined.  The  strain 


1219 


and  kinetic  energy  result  from  the  deformation  produced  by  the 
applied  strain  which  is  induced  by  exciting  the  piezoelectric  element. 
The  deformations  can  be  expressed  by  the  combination  of  the  midplane 
displacement  and  the  deformations  resulting  from  the  bending  of  the 
plate. 


Strain  Energy 

The  strain  energy  of  the  plate  and  piezoelectric  material  can  be 
calculated  by 

U  =  +T;«yY,<y)dVp  +T^T„)dVp.  (2) 

where  8  is  the  inplane  direct  strain,  a  is  the  inplane  direct  stress,  t  is 
the  inplane  shear  stress  and  y  is  shear  strain.  dV  shows  volume 
differential  and  indices  p  and  pe  refer  to  the  plate  and  piezoelectric 
actuator,  respectively.  The  strains  8^  ,  8^  and  y  can  be  shown  to  be 

For  the  Plate : 

du 

dv  d  (3) 

_du  dv  d^w 

^  dy  dx  dxdy 

For  the  Piezoelectric  actuator : 


Ev  = 


du 

a 

^~'dx~ 

^  a?* 

Sv 

a 

du  av 
«  ■  _  1  ^ 

dy  dx 

'"33=S,-H'^33 


a 


dxdy  ^ 


(4) 


where  _  refers  to  the  strains  due  to  the  deformation.  The  stresses  , 
Gy  ,  y  can  be  expressed  as 


1220 


For  the  Plate : 


E„ 

1 

0 

‘e.  ' 

1 

0 

y.vy. 

0 

0 

r.y_ 

2  . 

(5) 


For  the  Piezoelectric  actuator : 


' 

0^. 

II 

1 

0 

Sx-H^33 

1 

0 

r.y_ 

0 

0 

l-v 

2  . 

.  Is 

(6) 


where  E  is  Young  modulus  and  v  is  Poisson's  ratio  for  the  assumed 
isotropic  material.  Substituting  equations  3,  4,  5  and  6  into  2  yields  the 
strain  energy  of  the  plate  and  piezoelectric  actuator. 


^  S,  +  sj  +  1  ( 1  -  Dp)  dVp 
-  '"''33 )'  +  2Dpp(  e,  -  HV33 ){  E,  -  M.V33  )^  +  ^  1  ciVpp. 

•'Vp,  ^  _  u  —  -  tC 


(7) 


Kinetic  Energy 

To  obtain  the  kinetic  energy,  the  velocity  components  in  x,  y  and 
z  directions  are  needed.  The  velocity  components  can  be  calculated  by 
differentiating  the  displacement  components  which  are 

dw 

u—  u—  z— 
dx 

dw 

(8) 

dy 

w=w. 

Differentiating  equations  8  yields 


1221 


dw 

u=  u-  z  — 
ax 

dw 

v=  V- z — 

dy 

w 


(9) 


where  u  ,  y  and  w  are  the  velocity  components  in  the  x,  y  and  z 
directions  respectively.  Using  these  velocity  components,  the  kinetic 
energies  of  the  plate  and  piezoelectric  actuator  are  obtained  as 


dw 


P  rw^  +  (u-z^f +  (v-z^^]dV^ 


9x 


ay' 


(10) 


where  p  is  the  mass  density. 


Equation  of  Motion 

The  static  or  dynamic  response  of  the  plate  excited  by  the 
piezoelectric  actuator  can  be  calculated  by  substituting  the  strain  and 
kinetic  energy  into  Lagrange's  equation 


d  dT  dT  ^  dU 
dt  dq.  dq,"^  dq. 


(11) 


where  q^  represents  the  ith  generalised  coordinate  and  is  the  ith 
generalised  force.  As  there  are  no  external  forces  (the  force  applied  by 
5\e  piezoelectric  element  is  included  as  an  applied  strain)  or  gyroscopic 
terms  and  there  is  no  added  damping,  Lagrange's  equation  reduces  to  : 


dt  dq,  ^  dq, 


(12) 


Now  the  equation  of  motion  can  be  obtained  by  using  the  expression 
obtained  for  the  strain  and  kinetic  energy,  and  the  assumed  shape 
functions  for  flexural  and  longitudinal  motion 


1222 


u(x,y,t)~ 

v(x,y,t)={i,(x,y)y 

w(x,  y,  t)  =  {(|)(j:,  y)}^  {?(?)}. 


(13) 


Here  y  ,  ^  and  ([)  are  the  assumed  displacement  shape  and  h  ,  f  and  g 
are  generalised  coordinate  of  the  plate  response  m  x,  y  and  z 
directions.  Using  the  shape  functions  expressed  in  equations  13, 
substituting  equations  7  and  10  into  equation  12,  and  including 
Rayleigh  damping  yields  the  equation  of  motion  of  the  plate  in  the 
form 


where  M,  C  and  K  are  mass,  damping  and  stiffness  matrices  and  P  is 
the  voltage-to-force  transformation  vector.  Vector  q  represents  the 
plate  response  modal  amplitudes  and  V  is  the  applied  voltage. 

State-Space  Equations 

A  model  of  a  structure  found  via  finite  element  or  Rayliegh-Ritz 
methods  results  in  second-order  differential  equations  of  the  form 


[M]{9}+[C]{?}+[if]{?}={P}V  (15) 

Choosing  state  variables  x^_  =  q  and  =  i, ,  equation  1  may  be 

reduced  to  a  state-space  representation  as  follows  : 

q  =  x  =X2 

/  /  /  (16) 
q  =  X2=  ~M-^Kq  -  M~'Cq  +  M~'PV. 


Equations  15  can  then  be  rewritten  as 


r  0  /I 

0 

1 

1 

1 

1 

A. 

+ 

^2. 

.^2_ 

[,]  =  [/  0] 


(17) 


where  [  ],  {  }  are  ignored.  It  should  be  noted  that  the  vector  q  must  be 
multiplied  by  the  shape  fimctions  to  produce  the  actual  displacement. 


1223 


Results 

The  model  was  used  to  investigate  the  response  of  a  simply 
supported  plate.  In  order  to  maintain  symmetry  of  the  geometric 
structure  a  piezoelectric  actuator  is  assumed  to  be  bonded  to  both  the 
top  and  bottom  surfaces  of  the  plate.  So  The  symmetry  of  the  elements 
causes  no  extension  of  the  plate  midplane  and  the  plate  deforms  in 
pure  bending.  In  this  case  the  shape  functions  are  assumed  to  be  : 

\{/(x,y)  =  0 

^(x,y)  =  0  (18) 

41  ( X,  y )  =  jinf  x)sin(  ^  y) 

A- 

where  m  and  n  are  the  number  of  half  waves  in  the  x  and  y  directions. 

The  properties  of  the  plate  are  given  in  Table  1  and  its 
dimensions  are  4  =  Q38m  ,  4,  =  Q30/w  and  4=i.5876m/K.  Tables  2  and  3 
show  the  natural  frequencies  of  the  bare  plate  obtained  by  the  thin 
plate  theory  and  the  RR  model,  respectively.  Since  the  shape  functions 
used  in  this  example  express  the  exact  shape  of  the  simply  supported 
plate,  the  natural  frequencies  included  in  Tables  2  and  3  are  very  close. 

In  order  to  excite  the  plate,  a  piezoelectric  actuator  with 
configuration  x,  =  0.32  Xj  =  0.36  m,  =  0.04  OT  and  =  026  m  is  used  above 
and  below.  The  natural  frequencies  of  the  plate  after  bonding  the 
piezoelectric  actuator  to  the  surface  are  given  in  Table  4.  The  results 
show  an  increase  in  natural  frequencies,  showing  that  the  added 
stifness  is  more  important  than  the  added  inertia. 

Table  1 :  Properties  of  the  plate 

207  7870  .292 


Table  2 :  Plate  natural  frequencies  (rad  /  s) ,  Thin  Plate  Theory 


1 

2 

3 

4 

1 

437.5 

1246.0 

2593.5 

4480.0 

2 

941.4 

1749.9 

3097.4 

4983.9 

3 

1781.2 

2589.7 

3937.2 

5823.7 

4 

2957.0 

3765.5 

5113.0 

6999.5 

1224 


Table  3  :  Plate  natural  frequencies  (rad  /  s) ,  RR  Model 


/n 

1 

J 

2 

3 

4 

1 

437.5 

1245.9 

2593.2 

4479.1 

2 

941.3 

1749.7 

3097.0 

4982.8 

3 

1781.1 

2589.4 

3936.6 

5822.3 

4 

2956.7 

3764.9 

5111.9 

6997.4 

Table  4  :  Plate-Piezo  natural  frequencies  (rad  /  s) 


/n 

1 

2 

3 

4 

1 

444.0 

1257.1 

2611.3 

4502.8 

2 

957.2 

1775.6 

3182.1 

5076.7 

3 

1854.6 

2642.5 

4072.3 

6029.6 

4 

3076.6 

3933.8 

5224.6 

7277.2 

Initially,  the  piezoelectric  actuator  was  excited  by  a  constant  DC 
voltage.  The  result  of  this  action  is  shown  in  figure  2  which  shows  the 
dominant  out  of  plane  displacement  around  the  location  of  the 
piezoelectric  actuator  bonded  to  the  surface  of  the  plate.  To  show  the 
modes  of  vibration,  the  piezoelectric  actuator  was  excited  by  a  voltage 
with  frequencies  near  to  the  natural  frequencies  of  modes  (2,2)  and 
(1,3).  The  response  in  figures  3 , 4  show  that  the  piezoelectric  actuator 
excited  both  of  these  modes. 


Displacement  Distribution  (X-Y) 


0  0 


Figure  2.  Static  Displacement 


1225 


Displacement  Distribution  (X-Y) 


0  0 


Figure  3.  Vibration  of  the  plate,  mode  (2,2) 


Displacement  Distribution  (X-Y) 


0  0 


Figure  4.  Vibration  of  the  plate,  mode  (3,1) 

The  frequency  response  of  tihe  plate  at  the  centre  was  obtained 
by  exciting  the  piezoelectric  actuator  at  a  range  of  frequencies  between 
0  and  4000  rad/s.  Figure  5  shows  the  frequency  response  of  the  plate 
at  its  centre.  The  frequency  response  of  the  plate  at  y  =  0.5L^  along  the 
x-direction  is  shown  in  figure  6.  It  can  be  seen  that  the  amplitude  of 
vibration  of  some  modes  are  very  high  compared  to  that  of  the  other 
modes.  Special  attention  must  be  given  for  the  suppression  of  vibration 
of  these  modes. 


1226 


Frequency  Response  of  the  Plate 


Figure  5.  Frequency  response  of  the  plate  at  the  centre 


Frequency  Response  of  the  Plate 


4000 


Figure  6.  Frequency  response  of  the  plate  along  x-direction 

The  results  show  that  it  is  possible  to  predict  the  frequency 
response  of  a  plate  when  it  is  excited  by  a  patch  of  piezoelectric 
material.  Consequently,  a  sensor  model  can  be  also  added  to  the  model 
and  a  signal  proportional  to  velocity  fed  back  to  the  piezoelectric 
patch.  As  a  result,  the  closed  loop  frequency  response  of  the  plate  can 
be  obtained  theoretically  which  is  very  important  for  active  vibration 
control  studies.  This  also  permits  the  investigation  of  the  optimal 


1227 


location  of  the  actuators  and  the  study  of  control  algorithms  for  the 
best  possible  vibration  suppression  before  using  any  costly 
experimental  equipment. 

For  this  purpose,  two  patches  of  piezoelectric  (lOcmxScm  and 
5cmx4cm),  whose  specifications  are  listed  in  table  5,  were  bonded  to 
the  surface  of  the  plate  in  different  locations  and  then  the  plate  was 
excited  by  a  point  force  marked  by  "D"  in  the  figure  7.  In  figure  7  the 
dash  lines  are  showing  the  nodal  lines  of  a  simply  supported  plate  up 
to  mode  (3, 3). 


Table  5 :  Properties  of  the  actuator 
(mm)  EJxlO''^N/m')pJkg/m')  dJxlO'^'mfv)  ^ 

.2  6.25  7700  -180  .3 


An  actuator  is  most  effective  for  control  of  a  particular  mode  if 
the  sign  of  the  strain  due  to  the  modal  deflection  shape  is  the  same 
over  die  entire  actuator.  Consequently,  as  can  be  seen  from  figure  7, 
the  actuators  are  placed  between  the  nodal  lines  and  at  the  points  of 
maximum  curvature  in  order  to  obtain  good  damping  effect  on  the 
modes  of  interest.  Then  two  accelerometers  were  located  at  the  center 
of  the  location  of  the  actuators,  marked  by  "S"  in  figure  7,  in  order  to 
have  collocated  sensor-actuators.  The  signals  obtained  by  the 
accelerometers  are  integrated  and  fed  back  to  the  actuators  separately. 
Therefore  rate  feedback  was  used  in  this  configuration.  This  leads  to 
the  feedback  control  law 


V  =  kq 


(19) 


where  k  is  an  amplification  factor  or  feedback  gain.  Substituting 
equation  19  into  equation  17  the  closed-loop  state-space  representation 
of  the  system  can  then  be  obtained  as 


0 

-M-‘K 


_ J 

I — 

1 

M-^(C-kP)_ 

[^]=[/  0] 


(20) 


1228 


Then  the  effects  of  the  actuators  on  vibration  suppression  were 
investigated.  At  first,  only  the  actuator  near  to  the  center  of  the  plate 
was  used  to  suppress  the  vibration.  The  effect  of  this  is  shown  in  figure 
8.  As  can  be  seen,  damping  in  some  modes  are  improved  and  some 
modes  are  untouched. 

The  second  actuator  was  added  to  the  model  to  see  its  effect  on 
modes  of  vibration. 


Figure  7.  Plate  with  Bonded  Piezoelectric  Actuators 


Frequency  Response  of  the  Plate 


Figure  8.  Open  and  Closed  loop  Frequency  Response  of  the  plate 

solid  line ;  open  loop,  dash  line  :  closed  loop 


1229 


Frequency  Response  of  the  Plate 


-50 


_2qqI - 1 - 1 - 1 - 1 - 1 - 1 — — - 1 - 1 

0  500  1000  1500  2000  2500  3000  3500  4000 

Frequency  (rad/s) 

Figure  9.  Open  and  Closed  Loop  Frequency  Response  of  The  Plate 
solid  line :  open  loop,  dash  line  :  closed  loop 


The  open  and  closed  loop  frequency  response  of  the  plate  when  excited 
by  the  point  force  and  controlled  by  two  actuators  is  shown  in  figure  9. 
As  can  be  seen,  significant  vibration  suppression  was  obtained  in  both 
lower  and  higher  modes.  Also,  it  shows  that  the  place  of  actuators  was 
successfully  chosen.  This  analysis  showed  that  obtaining  reasonable 
but  not  necessarily  optimal  placement  of  actuators  in  structures  is  very 
important  in  order  to  obtain  a  high  level  of  damping  in  the  modes  of 
interest.  Obviously,  bonding  more  than  one  piezoelectric  actuator  in 
suitable  locations  helps  to  successfully  suppress  vibration  of  the  plate. 

Conclusions 

A  model  of  an  active  structure  is  fundamental  to  the  design  of 
control  strategies.  It  can  be  used  to  analyse  the  system  and  investigate 
optimal  control  strategies  without  using  costly  experimental 
equipment. 

A  Rayleigh-Ritz  model  has  been  developed  to  analyse  the  behaviour 
of  a  thin  plate  excited  by  a  patch  of  piezoelectric  material.  The  model 
has  been  used  for  a  simply  supported  plate.  It  has  been  shown  that  the 
model  can  predict  natural  frequencies  of  the  plate  alone  very 
accurately.  The  obtained  mode  shapes  also  correspond  to  the  actual 
mode  shapes.  The  frequency  response  of  the  plate  can  be  obtained  to 
show  the  suitability  of  the  model  for  control  design  studies.  This  study 
allowed  the  behaviour  of  the  system  in  open  and  closed  loop  form  for 
active  vibration  control  purposes  to  be  investigated.  Two  piezoelectric 
actuators  were  used  to  investigate  their  effectiveness  on  vibration 


1230 


suppression  of  the  plate.  The  analysis  showed  that  the  location  of  two 
actuators  was  very  important  to  increase  the  level  of  damping  in  both 
lower  and  higher  frequency  modes.  The  future  work  will  be  to  extend 
the  model  to  analyse  a  plate  with  more  than  two  patches  of 
piezoelectric  material  with  optimal  configuration,  obtained  by 
controllability  theory,  and  independent  controller  for  vibration 
suppression,  and  experimental  verification  of  the  theoretical  analysis. 


References 

1.  Crawley,  E.  F.  and  de  Luis,  J.,  Use  of  Piezoelectric  Actuators  as 
Elements  of  Intelligent  Structures,  AlAA  Journal,  Vol.  25,  No.  10, 
1987,  p.  1373. 

2.  Dosch,  J.,  Leo,  D.  and  Inman,  D.,  Modelling  and  control  for 
Vibration  Suppression  of  a  Flexible  Active  structure,  AIAA  Journal 
of  Guidance,  Control  and  Dynamics,  Vol.  18,  No.  2, 1995,  p.340. 

3.  Bailey,  T.  and  Hubbard  J.  E.  Jr.,  Distributed  Piezoelectric  Polymer 
Active  Vibration  Control  of  a  Cantilever  Beam,  AIAA  Journal  of 
Guidance,  Control  and  Dynamics,  Vol.  8,  No.  4, 1985,  p.605. 

4.  Dimitriadis,  E.  K.,  Fuller,  C.  R.,  Rogers  C.  A.,  Piezoelectric 
Actuators  for  Distributed  Vibration  Excitation  of  Thin  Plates, 
Journal  of  Vibration  and  Acoustics,  Vol.  113,  No.  1, 1991,  p.  100. 

5.  Azvine,  B.,  Tomlinson,  G.  R.  and  Wynne,  R.  J.,  Use  of  Active 
Constrained  Layer  Damping  for  Controlling  Resonant,  Journal  of 
Smart  Materials  and  Structures,  No.  4, 1995. 

6.  Rongong,  J.  A.,  Wright,  J.  R.,  Wynne,  R.  J.  and  Tomlinson,  G.  R., 
Modelling  of  a  Hybrid  Constrained  Layer/Piezoceramic  Approach 
to  Active  Damping,  Journal  of  Vibration  and  Acoustics,  To  appear. 

7.  Sadri,  A.  M.,  Wynne,  R.  J.  and  Cherry,  A.  S.,  Modelling  and 
Control  of  Active  Damping  for  Vibration  Suppression,  UKACC 
International  Conference  on  Control'  96, 2-5  September  1996. 

8.  Bathe,  K,,  Finite  Element  Procedures  in  Engineering  Analysis,  Prentice- 
Hall,  Inc.,  1982. 

9.  Blevins,  R.  D.,  Formulas  for  Natural  Frequency  and  Mode  Shapes,  Van 
Nostrand  Remhold,  NY  1979. 

10.  Thomson,  W.  T.,  Theory  of  Vibration  with  Applications,  Prentice-Hall, 
Inc.,  1988. 


1231 


Active  control  of  sound  transmission  into  a  rectangular 
enclosure  using  both  structural  and  acoustic  actuators 


S.M.  Kim  and  MJ.  Brennan 

ISVR,  University  of  Southampton,  Highfieid,  Southampton,  S017  IBJ,  UK 


ABSTRACT 

This  paper  presents  an  analytical  investigation  into  the  active  control  of  sound 
transmission  in  a  ‘weakly  coupled’  structural-acoustic  system.  The  system 
under  consideration  is  a  rectangular  enclosure  having  one  flexible  plate 
through  which  external  noise  is  transmitted.  Three  active  control  systems 
classified  by  the  type  of  actuators  are  discussed.  They  are;  i)  a  single  force 
actuator,  ii)  a  single  acoustic  piston  source,  and  iii)  simultaneous  use  of  both 
the  force  actuator  and  the  acoustic  piston  source.  For  all  three  control  systems 
the  acoustic  potential  energy  inside  the  enclosure  is  adopted  as  the  cost 
function  to  minimise,  and  perfect  knowledge  of  the  acoustic  field  is  assumed. 
The  results  obtained  demonstrate  that  a  single  point  force  actuator  is  effective 
in  controlling  well  separated  plate-controlled  modes,  whereas,  a  single 
acoustic  piston  source  is  effective  in  controlling  well  separated  cavity- 
controlled  modes  provided  the  discrete  actuators  are  properly  located.  Using 
the  hybrid  approach  with  both  structural  and  acoustic  actuators,  improved 
control  effects  on  the  plate  vibration  together  with  a  further  reduction  in 
transmitted  noise  and  reduced  control  effort  can  be  achieved.  Because  the 
acoustic  behaviour  is  governed  by  both  plate  and  cavity-controlled  modes  in  a 
‘weakly  coupled’  structural-acoustic  system,  the  hybrid  approach  is  desirable 
in  this  system. 


1.  INTRODUCTION 

Analytical  studies  of  vibro-acoustic  systems  have  been  conducted  by  many 
investigators  to  achieve  physical  insight  so  that  effective  active  control 
systems  can  be  designed.  It  is  well  established  that  a  single  point  force  actuator 
and  a  single  acoustic  piston  source  can  be  used  to  control  well  separated 
vibration  modes  in  structures  and  well  separated  acoustic  modes  in  cavities, 
respectively,  provided  that  the  actuators  are  positioned  to  excite  these 
modes  [1,2].  Active  control  is  also  applied  to  structural-acoustic  coupled 
systems  for  example,  the  control  of  sound  radiation  from  a  piate[3-6]  and  the 
sound  transmission  into  a  rectangular  enclosure[7-8].  Meirovitch  and 


1233 


Thangjitham[6],  who  discussed  the  active  control  of  sound  radiation  from  a 
plate,  concluded  that  more  control  actuators  resulted  in  better  control  effects. 
Pan  et  al[Z]  used  a  point  force  actuator  to  control  sound  transmission  into  an 
enclosure,  and  discussed  the  control  mechanism  in  terms  of  plate  and  cavity- 
controlled  modes. 

This  paper  is  concerned  with  the  active  control  of  sound  transmission  into  a 
‘weakly  coupled’  structural-acoustic  system  using  both  structural  and  acoustic 
actuators.  After  a  general  formulation  of  active  control  theory  for  structural- 
acoustic  coupled  systems,  it  is  applied  to  a  rectangular  enclosure  having  one 
flexible  plate  through  which  external  noise  is  transmitted.  Three  active  control 
systems  classified  by  the  type  of  actuators  are  compared  using  computer 
simulations.  They  are;  i)  a  single  force  actuator,  ii)  a  single  acoustic  piston 
source,  and  iii)  simultaneous  use  of  both  the  force  actuator  and  the  acoustic 
piston  source.  For  all  three  control  systems  the  acoustic  potential  energy  inside 
the  enclosure  is  adopted  as  the  cost  function  to  minimise,  and  perfect 
knowledge  of  the  acoustic  field  is  assumed.  The  effects  of  each  system  are 
discussed  and  compared,  and  a  desirable  control  system  is  suggested. 


2.  THEORY 

2.1  Assumptions  and  co-ordinate  systems 

Consider  an  arbitrary  shaped  enclosure  surrounded  by  a  flexible  structure  and  a 
acoustically  rigid  wall  as  shown  in  Figure  1.  A  plane  wave  is  assumed  to  be 
incident  on  the  flexible  structure,  and  wave  interference  outside  the  enclosure 
between  the  incident  and  radiated  waves  by  structural  vibration  is  neglected. 
Three  separate  sets  of  co-ordinates  systems  are  used;  Co-ordinate  x  is  used  for 
the  acoustic  field  in  the  cavity,  co-ordinate  y  is  used  for  the  vibration  of  the 
structure,  and  co-ordinate  r  is  used  for  the  sound  field  outside  the  enclosure. 
The  cavity  acoustic  field  and  the  flexible  structure  are  governed  by  the  linear 
Helmholtz  equation  and  the  isotropic  thin  plate  theory[9],  respectively.  The 
sign  of  the  force  distribution  function  and  normal  vibration  velocity  are  set  to 
be  positive  when  they  direct  inward  to  the  cavity  so  that  the  structural 
contribution  to  acoustic  pressure  has  the  same  sign  as  the  acoustic  source 
contribution  to  acoustic  pressure. 

Weak  coupling  rather  than  full  coupling  is  assumed  between  the  structural 
vibration  system  and  the  cavity  acoustic  system.  Thus,  the  acoustic  reaction 
force  on  the  strucural  vibration  under  structural  excitation  and  the  structural 
induced  source  effect  on  the  cavity  acoustic  field  under  acoustic  excitation  is 
neglected.  This  assumption  is  generally  accepted  when  the  enclosure  consists 
of  a  heavy  structure  and  a  big  volume  cavity.  It  is  also  assumed  that  the 
coupled  response  of  the  system  can  be  described  by  finites  summations  of  the 


1234 


uncoupled  acoustic  and  structural  modes.  The  uncoupled  modes  are  the  rigid- 
walled  acoustic  modes  of  the  cavity  and  the  in  vacuo  structural  modes  of  the 
structure.  The  acoustic  pressure  and  structural  vibration  velocity  normal  to  the 
vibrating  surface  are  chosen  to  represent  the  responses  of  the  coupled  system. 

2.2  Structural-acoustic  coupled  response 

The  acoustic  potential  energy  in  the  cavity  is  adopted  as  the  cost  function  for 
the  global  sound  control,  which  is  given  by  [2] 

where,  and  Co  respectively  denote  the  density  and  the  speed  of  sound  in  air, 
and  /?(x,ft))  is  the  sound  pressure  inside  the  enclosure. 

The  vibrational  kinetic  energy  of  the  flexible  structure,  which  will  be  used  to 
judge  the  control  effect  on  structural  vibration,  is  given  by[l] 

where,  p.,  is  the  density  of  the  plate  material,  h  is  the  thickness  of  the  plate. 

If  the  acoustic  pressure  and  the  structural  vibration  are  assumed  to  be 
described  by  a  summation  of  N  and  M  modes,  respectively,  then  the  acoustic 
pressure  at  position  x  inside  the  enclosure  and  the  structural  vibration  velocity 
at  position  y  are  given  by 

N 

P(X,(B)  = 

n=I 

M 

ni=l 

where,  the  N  length  column  vectors  ^  and  a  consist  of  the  array  of  uncoupled 
acoustic  mode  shape  functions  and  the  complex  amplitude  of  the 

acoustic  pressure  modes  a^^(co)  respectively.  Likewise  the  M  length  column 

vectors  O  and  b  consist  of  the  array  of  uncoupled  vibration  mode  shape 
functions  0n,(y)  and  the  complex  amplitude  of  the  vibration  velocity  modes 

respectively. 

The  mode  shape  functions  \f/^{x)  and  (l)Jy)  satisfy  the  orthogonal  property 
in  each  uncoupled  system,  and  can  be  normalised  as  follows. 

V  =  lwlMdV  (5) 

S,=lfyy)dS  (6) 


1235 


where,  V  and  S/  are  the  volume  of  the  enclosure  and  the  area  of  the  flexible 
structure,  respectively.  Since  mode  shape  functions  are  normalised  as  given  by 
(Eq.  5),  the  acoustic  potential  energy  can  be  written  as 

£  =_L^a“a  (7) 

'  4p„c^ 

Similarly  from  (Eq.  2)  and  (Eq.  6),  the  vibrational  kinetic  energy  can  be 
written  as 

£  =££^b«b  (8) 

2 

Where  superscript  H  denotes  the  Hermitian  transpose. 

For  the  global  control  of  sound  transmission,  it  is  required  to  have  knowledge 
of  the  complex  amplitude  of  acoustic  pressure  vector  a  for  various  excitations. 

The  complex  amplitude  of  the  n-th  acoustic  mode  under  structural  and 
acoustic  excitation  is  given  by[9, 10] 

^1.  (^)  =  4  (^)(^  (x)‘y(x,  co)dv + i//'„  (yMy,  j  (9) 

where,  s(x,Ci))  denotes  the  acoustic  source  strength  density  function  in  the 
cavity  volume  V”,  and  w(y,6))  denotes  the  normal  velocity  of  the  surrounding 
flexible  structure  on  surface  5/-.  The  two  integrals  inside  the  brackets  represent 
the  nth  acoustic  modal  source  strength  contributed  from  s(x,Q))  and  u(y,co) , 
respectively.  The  acoustic  mode  resonance  term  An(fi))  is  given  by 

A,.(co)  =  - - -  (10) 

o}--Q)-+j2C„<a„(o 

where  a)„and  C„  are  the  natural  frequency  and  damping  ratio  of  the  nth 
acoustic  mode,  respectively. 

Substituting  (Eq.  4)  into  (Eq.  9)  and  introducing  the  modal  source  strength 
q,:  =  j  ¥n  X.  0))dV ,  then  we  get 


2  /  M 

a„ (®)  =  A, (®)  (®)  +  S O',,™  •  b,„((0) 

y  V  Hi=i 


where,  C„,,„  represents  the  geometric  coupling  relationship  between  the 
uncoupled  structural  and  acoustic  mode  shape  functions  on  the  surface  of  the 
vibrating  structure  Sf  and  is  given  by[l  1] 

c,,,„  =  lvJy)<t>Jy’0>)ds  (12) 

If  we  use  L  independent  acoustic  control  sources,  can  be  written  as 

/=!  \l  /=! 


1236 


where  = - j  W ^  control  source  strength  q,^i{co) 

having  an  area  of  Scjj  is  defined  at  Xc,i. 

Thus,  the  complex  amplitude  of  acoustic  modal  pressure  vector  a  can  be 
expressed  as 

a  =  Z.(D,q,+Cb)  (14) 

2 

whereZa=  -^A. 

The  matrix  A  is  a  (NxN)  diagonal  matrix  in  which  each  (n,n)  diagonal  term 
consists  of  A„,  the  (NxM)  matrix  C  is  the  structural-acoustic  mode  shape 
coupling  matrix,  the  (NxL)  matrix  Dq  determines  coupling  between  the  L 
acoustic  source  locations  and  the  N  acoustic  modes,  the  L  length  vector  (jc  is 
the  complex  strength  vector  of  acoustic  control  sources,  and  b  is  the  complex 
vibrational  modal  amplitude  vector.  The  {NxN)  diagonal  matrix  Za  can  be 
defined  as  the  uncoupled  acoustic  modal  impedance  matrix  which  determines 
the  relationship  between  the  acoustic  source  excitation  and  the  resultant 
acoustic  pressure  in  modal  co-ordinates  of  the  uncoupled  acoustic  system. 
Generally  the  impedance  matrix  is  symmetric  but  non-diagonal  in  physical  co¬ 
ordinates,  however  the  uncoupled  modal  impedance  matrix  is  diagonal 
because  of  the  orthogonal  property  of  uncoupled  modes  in  modal  co-ordinates. 


Since  the  flexible  structure  in  Figure  1  is  assumed  to  be  governed  by  the 
isotropic  thin  plate  theory,  the  complex  vibrational  velocity  amplitude  of  the 
mth  mode  can  be  expressed  as[10] 

=  +  (15) 

p,hSf  ^ 

where,  again  p.v  is  the  density  of  the  plate  material,  h  is  the  thickness  of  the 
plate,  is  the  area  of  flexible  structure.  Inside  the  integral  /(y,ty) ,  p""\'^.(0) , 
and  p‘"'(y,ty)  denote  the  force  distribution  function,  and  the  exterior  and 
interior  sides  of  acoustic  pressure  distribution  on  the  surface  5/ ,  respectively. 
Because  of  the  sign  convention  used,  there  is  a  minus  sign  in  front  of 
(y,  (O) .  The  structural  mode  resonance  term  Bm((0)  can  be  expressed  as 


-(0^+j2^„^0),„co 


where  co^  and  Cm  natural  frequency  and  the  damping  ratio  of  m-th 


mode,  respectively.  Substituting  (Eq.  3)  into  (Eq.  15),  then  we  get 

b,niO})=——B,„ico)\  g,,,„ico)+  g^,„i(o)-'£cl„-a,(co)  (17) 

p,hSf  \  „=i  ; 


1237 


where,  L  =  L 

•  JSj-  •'■i/ 

and  Cl^  =  C,„..  ■ 

If  we  use  K  independent  point  force  actuators,  the  m-th  mode  generalised  force 
due  to  control  forces,  gc.m,  can  be  written  by 

=  XI  <P,n(y)S(y-ycjc)dsf,j,ico)  =  X^/.».t/<.-.t(®) 

k=\  ^  k=] 

(18) 

where  0,„(y)5(y-y.,t)rfS,  and  the  k-th  control  point  force  l,{co) 

JSj 

is  located  at  y^- it. 

Thus  the  modal  vibrational  amplitude  vector  b  can  be  expressed  as 

b  =  Y,(g,+Dff,-C’'a)  (19) 

where  Ys  =  — ^ —  B. 

P..hSjr 

The  matrix  B  is  a  (MxM)  diagonal  matrix  in  which  each  (m,m)  diagonal  term 
consists  of  B,„,  is  the  transpose  matrix  of  C,  the  (NxK)  matrix  Df 
determines  coupling  between  the  K  point  force  locations  and  the  M  structural 
modes,  gp  is  the  generalised  modal  force  vector  due  to  the  primary  plane  wave 
excitation,  the  K  length  vector  fc  is  the  complex  vector  of  structural  control 
point  forces,  and  a  is  the  complex  acoustic  modal  amplitude  vector.  The 
(MxM)  diagonal  matrix  Ys  can  be  defined  as  the  uncoupled  structural  modal 
mobility  matrix  which  determines  the  relationship  between  structural 
excitation  and  the  resultant  structural  velocity  response  in  modal  co-ordinates 
of  the  uncoupled  structural  system.  As  with  the  uncoupled  acoustic  impedance 
matrix  Za,  note  that  Ys  is  a  diagonal  matrix. 


From  (Eq.  14)  and  (Eq.  19),  we  get 


a  =  (l  +  Z,CY,C’^)''z,(D,q,+CY,g,+CWc) 

(20) 

b  =  (I  +  Y,C’'Z,C)''  Y,(gp  +  Dff,  -  C^Z.D^q,) 

(21) 

Since  weakly  coupling  is  assumed  i.e.  =  0  and  Y^C^Z^C  =  0  , 

then 

we  get 

a  =  Z,(D,q,+CY,g,+CY,D,f.) 

(22) 

b  =  X(g,+D,f,-C%D,q,) 

(23) 

Although  the  formulation  developed  above  covers  fully  coupled  systems,  weak 
coupling  is  assumed  hereafter  for  the  convenience  of  analysis. 


1238 


In  order  to  minimise  the  sound  transmission  into  the  cavity,  two  kinds  of 
actuators  are  used:  a  single  point  force  actuator  for  controlling  the  structural 
vibration  of  the  plate  and  a  single  rectangular  type  acoustic  piston  source  for 
controlling  the  cavity  acoustic  pressure.  The  rectangular  piston  source  is 
centred  at  (1.85,0.15,0)  with  the  area  of  0.15m  by  0.15m,  This  location  was 
chosen  because  the  sound  pressure  of  each  mode  in  a  rectangular  cavity  is  a 
maximum  at  the  corners,  and  thus  the  control  source  is  placed  away  from  the 
acoustic  nodal  planes  [2].  For  a  similar  reason,  the  point  force  actuator  is 
located  at  (9/20Li,  L2/2)  on  the  plate,  at  which  there  are  no  nodal  lines  within 
the  frequency  range  of  interest.  Table  2  shows  the  natural  frequencies  of  each 
uncoupled  systems  and  their  geometric  mode  shape  coupling  coefficients 
which  are  normalised  by  their  maximum  value.  Some  of  natural  frequencies 
which  are  not  excited  by  the  given  incident  angie((p  =  0°)  were  omitted.  The 
(m/,  m2)  and  («/,  112,  ns)  indicate  the  indices  of  the  m-th  plate  mode  and  the  n\h 
cavity  mode,  and  corresponding  the  uncoupled  natural  frequencies  of  the  plate 
and  the  cavity  are  listed.  A  total  15  structural  and  10  acoustic  modes  were  used 
for  the  analysis  under  300  Hz,  and  no  significant  difference  was  noticed  in 
simulations  with  more  modes. 


3.2  Active  minimisation  of  the  acoustic  potential  energy 

This  section  considers  an  analytical  investigation  into  the  active  control  of  the 
sound  transmission  into  the  rectangular  enclosure  in  Figure  2.  Three  active 
control  strategies  classified  by  the  type  of  actuators  are  considered.  They  are; 
i)  a  single  force  actuator,  ii)  a  single  acoustic  piston  source,  and  iii) 
simultaneous  use  of  both  the  force  actuator  and  the  acoustic  piston  source. 
Although  the  formulation  developed  in  this  paper  is  not  restricted  to  a  single 
actuator,  each  single  actuator  was  used  to  simplify  problems  so  that  the  control 
mechanisms  could  be  understood  and  effective  guidelines  for  practical 
implementation  could  be  established. 

3.2.1  control  using  a  single  force  actuator 

A  point  force  actuator  indicated  in  Figure  2  is  used  as  a  structural  actuator  and 
the  optimal  control  strength  of  the  point  force  actuator  can  be  calculated  using 
(Eq.  26).  Figure  3(a)  shows  the  acoustic  potential  energy  of  the  cavity  with 
and  without  the  control  force.  To  show  how  this  control  system  affects  the 
vibration  of  the  plate,  the  vibration  kinetic  energy  of  the  plate  obtained  from 
(Eq.  8)  is  also  plotted  in  Figure  3(b).  On  each  graph,  natural  frequencies  of  the 
plate  and  the  cavity  are  marked  and  ‘o’  at  the  frequencies,  respectively.  It 
can  be  seen  that  the  acoustic  response  of  uncontrolled  state  has  peaks  at  both 


1239 


plate  and  cavity  resonances,  and  the  vibration  response  of  uncontrolled  state  is 
governed  by  the  plate  resonances  only  because  of  ‘weak  coupling’. 

Examining  Figure  3(a,b)  it  can  be  seen  that  at  the  1st,  2nd,  4th,  and  5th  plate 
modes  corresponding  to  52  Hz,  64  Hz,  115  Hz,  and  154  Hz,  respectively  there 
is  a  large  reduction  of  the  acoustic  potential  energy.  This  is  because  the  sound 
field  at  these  frequencies  is  governed  by  the  plate  vibration  modes,  and  a 
single  structural  actuator  is  able  to  control  the  corresponding  vibration  mode  to 
minimise  sound  transmission. 

The  structural  actuator  reduces  sound  at  cavity-controlled  modes  as  well( 
especially  the  2nd  and  3rd  cavity  modes  corresponding  respectively  to  85  Hz 
and  170  Hz),  however  it  has  to  increase  plate  vibration  significantly.  It  shows 
that  minimisation  of  the  acoustic  potential  energy  does  not  always  bring  the 
reduction  of  structural  vibration,  and  vice  versa.  Since  a  cavity-controlled 
mode  is  generally  well  coupled  with  several  structural  modes,  a  single 
structural  actuator  is  not  able  to  deal  with  several  vibration  modes  because  of 
'control  spillover\A\.  This  is  the  reason  why  a  single  acoustic  piston  source 
used  in  the  next  section  was  introduced.  However,  it  is  clear  that  a  single 
point  force  actuator  is  effective  in  controlling  a  well  separated  plate-controlled 
mode  provided  the  actuator  is  not  located  close  to  the  nodal  line. 


3.2.2  control  using  a  single  piston  force  source 

A  single  acoustic  piston  source  indicated  in  Figure  2  is  used  for  controlling 
the  acoustic  sound  field  directly.  The  optimal  control  source  strength  of  the 
acoustic  piston  source  can  be  determined  using  (Eq.  26).  Figure  4  shows  the 
acoustic  potential  energy  of  the  cavity  and  the  vibrational  kinetic  energy  of  the 
plate  with  and  without  the  control  actuators. 

Since  a  plate-controlled  mode  is  generally  coupled  with  several  cavity  modes, 
the  control  effect  of  the  acoustic  source  is  not  effective  at  plate-controlled 
modes  (e.g.  52  Hz,  64  Hz,  115  Hz  etc.).  Whereas,  it  is  more  able  to  reduce 
transmitted  sound  at  the  cavity-controlled  modes  (  e.g.  0  Hz,  85  Hz,  170  Hz 
and  189  Hz)  than  the  structural  actuator.  As  can  be  noticed  from  (Eq.  22),  the 
external  incident  wave  and  force  excitation  have  the  same  sound  transmission 
mechanism,  which  is  vibrating  the  plate  and  transmitting  sound  through  the 
geometric  mode  shape  coupling  matrix  C.  Thus,  it  can  be  said  that  the 
structural  actuator  is  generally  effective  in  controlling  sound  transmission.  At 
cavity  controlled  modes,  however,  several  vibration  modes  are  coupled  with 
an  acoustic  mode.  It  means  that  a  single  acoustic  source  is  more  effective  than 
a  single  structural  actuator  since  a  single  actuator  is  generally  able  to  control 
only  one  mode.  From  the  results,  it  is  clear  that  a  single  acoustic  piston  source 
is  effective  in  controlling  well  separated  cavity-controlled  modes.  It  is 
interesting  that  there  is  not  much  difference  in  the  vibrational  kinetic  energy 


1240 


with  and  without  control  state.  It  means  that  the  acoustic  actuator  is  able  to 
reduce  sound  field  globally  without  increasing  plate  vibration. 


3.2.3  control  using  both  the  piston  source  and  the  structural  actuator 

In  this  section,  a  hybrid  approach,  simultaneous  use  of  both  the  point  force 
actuator  and  the  acoustic  piston  source,  is  applied.  The  optimal  strength  of  the 
force  actuator  and  the  piston  source  can  be  obtained  from  Eq.  (26).  Figure  5 
shows  the  acoustic  potential  energy  of  the  cavity  and  the  vibrational  kinetic 
energy  of  the  plate  with  and  without  the  control  actuators.  Even  at  the  cavity- 
controlled  modes,  it  can  be  seen  that  a  large  reduction  in  the  acoustic  potential 
energy  is  achieved  without  significantly  increasing  the  structural  vibration.  In 
general,  more  control  actuators  result  in  better  control  effects[6].  However,  the 
hybrid  approach  with  both  structural  and  acoustic  actuators  in  the  system  does 
not  merely  mean  an  increase  in  the  number  of  actuators.  As  demonstrated  in 
the  last  two  sections,  a  single  structural  actuator  is  effective  in  controlling 
well  separated  plate-controlled  modes  and  a  single  acoustic  actuator  is 
effective  for  controlling  well  separated  cavity-controlled  modes.  Since  the 
acoustic  response  is  governed  by  both  plate-controlled  and  cavity-controlled 
modes,  the  hybrid  control  approach  can  be  desirable  for  controlling  sound 
transmission  in  a  ‘weakly-coupled’  structural  acoustic  system. 

To  investigate  the  control  efforts  of  each  control  system,  the  amplitude  of  the 
force  actuator  and  the  source  strength  of  the  piston  source  are  plotted  in 
Figure  6.  There  is  a  large  decrease  of  the  force  amplitude  at  the  well 
separated  cavity-controlled  modes,  e.g.  85  Hz  and  170  Hz,  by  using  the  both 
actuators.  This  trend  can  also  be  seen  in  the  case  of  the  piston  source  strength, 
especially  at  the  1st  and  2nd  structural  natural  frequency  (52  Hz,  64  Hz).  By 
using  the  hybrid  approach,  simultaneous  use  of  both  actuators,  better  control 
effects  on  the  vibration  of  the  plate,  the  transmission  noise  reduction  and  the 
control  efforts  of  the  actuators  can  be  achieved. 


4.  Conclusion 

The  active  control  of  the  sound  transmission  into  a  ‘weakly  coupled’ 
structural-acoustic  system  has  been  considered.  The  results  obtained 
demonstrates  that  a  single  point  force  actuator  is  effective  in  controlling  well 
separated  plate-controlled  modes,  whereas,  a  single  acoustic  piston  source  is 
effective  in  controlling  well  separated  cavity-controlled  modes. 

By  using  the  hybrid  approach  with  both  structural  and  acoustic  actuators, 
improved  control  effects  on  the  plate  vibration,  further  reduction  in  sound 


1241 


transmission,  and  reduced  control  efforts  of  the  actuators  can  be  achieved. 

Since  the  acoustic  behaviour  is  governed  by  both  plate  and  cavity  resonances, 

the  hybrid  control  approach  can  be  desirable  in  controlling  sound  transmission 

in  a  ‘weakly  coupled’  structural-acoustic  system. 

References 

1.  C.R.  FULLER,  S.J.  ELLIOTT  and  P.A.  NELSONActive  control  of 
vibration,  Academic  Press  Limited,  1996 

2.  P.A.  NELSON  and  S.J.  ELLIOTT  Active  control  of  sound,  Academic 
Press  Limited,  1992 

3.  B.-T.  WANG,  C.R.  FULLER  and  K.  DIMITRIADIS  Active  control  of 
noise  transmission  through  rectangular  plates  using  multiple 
piezoelectric  or  point  force  actuators  Journal  of  the  Acoustical  Society 
of  America,  1991,  90(5),  2820-2830. 

4.  M.E.  JOHNSON  and  S.  J.  ELLIOTT  Active  control  of  sound  radiation 
using  volume  velocity  cancellation.  Journal  of  the  Acoustical  Society  of 
America,  1995,  98(4),  2174-2186. 

5.  C.R.  FULLER,  C.H.  HANSEN  and  S.D.  SNYDER  Active  control  of 
sound  radiation  from  a  vibrating  rectangular  panel  by  sound  sources 
and  vibration  inputs:  an  experimental  comparison,  Journal  of  Sound 
and  Vibration,  1991, 145(2),  195-215. 

6.  L.  MEIROVrrCH  and  S.  THANGJITHAM  Active  control  of  sound 
radiation  pressure,  Trans,  of  the  ASMS  Journal  of  Vibration  and 
Acoustics,  1990, 112,  237-244. 

7.  S.D.  SNYDER  and  N.  TANAKA  On  feedforward  active  control  of 
sound  and  vibration  using  vibration  error  signals,  Journal  of  the 
Acoustical  Society  of  America,  1993,  94(4),  2181-2193. 

8.  J.  PAN  C.H.  HANSEN  and  D.  A.  BIES  Active  control  of  noise 
transmission  through  a  panel  into  a  cavity  :  I.  analytical  study.  Journal 
of  the  Acoustical  Society  of  America,  1990,  87(5),  2098-2108. 

9.  P.M.  MORSE  and  K.U.  INGARD  Theoretical  Acoustics,  McGraw- 
Hill,  1968 

10.  E.H.  DOWELL,  G.F.  GORMAN  HI,  and  D.A.  SMITH 
Acoustoelasticity  :  general  theory,  acoustic  modes  and  forced  response 
to  sinusoidal  excitation,  including  comparisons  with  experiment. 
Journal  of  Sound  and  Vibration,  1977,  52(4),  519-542. 

11.  F.  FAHY  Sound  and  Structural  Vibration,  Radiation,  Transmission 
and  Response,  Academic  Press  Limited,  1985 


1242 


incident  plane  wave 


Figure  1  A  structural  acoustic  coupled  system  with  the  volume  V  and  its  flexible  boundary 
surface  S/. 


Figure  2  The  rectangular  enclosure  with  one  simply  supported  plate  on  the  surface  Sf  on 
which  external  plane  wave  is  incident  with  the  angles  of  (cp  =  0°)  and  (0=45°). 


Table  1  Material  properties 


Material 

Density 

Phase  speed 

Young’s 

Poisson’s 

Damping 

(kg/m^) 

(m/s) 

modulus  (N/m^) 

ratio  (v) 

ratio  (0 

Air 

1.21 

340 

- 

- 

0.01 

Steel 

7870 

- 

207x10® 

0.292 

0.01 

1243 


Table  2  The  natural  frequencies  and  geometric  mode  shape  coupling  coefficients  of  each 
uncoupled  system 


Order: 

Plate, 

1 

2 

3 

4 

■„  ,  5. 

7 

10 

■c 

’Type 

(1,1) 

(2,1):: 

0,1)  : 

(4,1) 

(5,1) 

(6,1) 

(7,1) 

Cavity^ 

'  Freq.- 

52  Hz 

64  Hz 

86Hz 

115  Hz 

154  Hz 

200  Hz 

;256Hz 

1 

(0,0,0) 

0  Hz 

0.71 

0 

0.24 

0 

0.14 

0 

O.IO 

2 

(1,0.0) 

85  Hz 

0 

0.67 

0 

0.27 

0 

0.17 

0 

3 

(2,0.0) 

170  Hz 

-0.33 

0 

0.60 

0 

0.24" 

0 

0.16 

4 

(0,0.1) 

189  Hz 

-1.00 

0 

-0.33 

0 

-0.20" 

0 

-0.14 

5 

(1,0,1) 

207.HZ 

0 

-0.94 

0 

-0.38 

0 

-0.24 

0 

6 

(2;0,1) 

254  Hz 

0.47 

0 

-0.85 

0 

-0.34 

0 

-0.22 

7 

(3,0,0) 

255  Hz 

0 

-0.40 

0 

0.57 

0 

0.22 

0 

1 oo  ISO  200  2SO 

Frequency 

(a)  the  acoustic  potential  energy  of  the  cavity 


TSO 

Frequency 


(b)  the  vibrational  kinetic  energy  of  the  plate(dB  ref .=  10  ^  J) 

Figure  3  Effects  of  minimising  the  acoustic  potential  energy  using  a  point  force  actuator  ( 
solid  line  :  without  control,  dashed  line  :  with  control  ),  where  **’  and  ‘o’  are  at  uncoupled 
plate  and  cavity  natural  frequencies,  respectively. 


301 


Frequency 

(a)  the  acoustic  potential  energy  of  the  cavity 


Figure  4  Effects  of  minimising  the  acoustic  potential  energy  using  an  acoustic  piston  source 
(  solid  line  ;  without  control,  dashed  line  :  with  control ),  where  and  ‘o’  are  at  uncoupled 
plate  and  cavity  natural  frequencies,  respectively. 


Figure  5  Effects  of  minimising  the  acoustic  potential  energy  using  both  a  point  force 
actuator  and  an  acoustic  piston  source  -  continued 


2 


Frequency 

(b)  the  vibrational  kinetic  energy  of  the  pIate(dB  ref =10‘^  J) 


Figure  5  Effects  of  minimising  the  acoustic  potential  energy  using  both  a  point  force 
actuator  and  an  acoustic  piston  source  (  solid  line  :  without  control,  dashed  line:  with  control 
),  where  and  ‘o’  are  at  uncoupled  plate  and  cavity  natural  frequencies,  respectively. 


Frequency 

(a)  the  strength  of  the  force  actuator 


Frequency 

(b)  the  strength  of  the  piston  source(  unit:  mVsec) 

Figure  6  Comparison  of  control  efforts  of  the  three  control  strategies;  using  each  actuator 
separately  (  solid  line  )  and  using  both  the  force  actuator  and  the  piston  source  (dashed  line  ) 
,  where  and  ‘o’  are  at  uncoupled  plate  and  cavity  natural  frequencies,  respectively. 


1246 


A  DISTRIBUTED  ACTUATOR  FOR  THE 
ACTIVE  CONTROL  OF  SOUND 
TRANSMISSION  THROUGH  A  PARTITION 


TJ.  Sutton,  M.E.  Johnson  and  S.J.  Elliott 

Institute  of  Sound  and  Vibration  Research 
University  of  Southampton,  Southampton  S017 IBJ 


ABSTRACT 

The  paper  considers  the  problem  of  active  control  of  soimd  transmission 
through  a  partition  using  a  single  distributed  actuator.  The  use  of  shaped, 
distributed  actuators  rather  than  point  sources  or  locally-acting  piezoceramic 
elements  offers  the  possibility  of  controlling  the  volume  velocity  of  a  plate 
without  giving  rise  to  control  spillover  and  avoids  an  increase  in  the  sound 
radiated  by  uncontrolled  structural  modes.  Specifically,  a  form  of  distributed 
piezoelectric  actuator  is  described  in  which  the  electrode  takes  the  form  of  a 
set  of  quadratic  strips  and  serves  to  apply  a  roughly  uniform  normal  force 
over  its  surface. 


INTRODUCTION 

The  strong  piezoelectric  properties  of  the  polymer  polyvinylidene  fluoride 
(PVDF)  were  discovered  in  1969  [1].  The  material  is  lightweight,  flexible, 
inexpensive  and  can  be  integrated  into  engineering  structures  for  strain 
sensing  and  to  apply  distributed  forces  and  moments  for  the  active  control  of 
vibration  and  sound  transmission.  Such  ^smarf  materials  offer  the  possibility 
of  providing  lightweight  sound-insulating  barriers  for  application  to  aircraft, 
ground-based  transport  and  in  buildings. 

Lee  [2]  has  set  out  the  underlying  theory  of  active  laminated  structures  in 
which  one  or  more  layers  of  flexible  piezoelectric  material  are  attached  to  a 
plate.  Practical  sensors  using  PVDF  material  have  been  implemented  by 
Clark  and  Fuller  [3],  Johnson  and  Elliott  [4-6],  and  others.  In  these  cases  thin 
PVDF  films  were  attached  to  the  structure  to  sense  integrated  strain  over  a 
defined  area.  In  [4]  for  example  a  distributed  sensor  was  developed  whose 
output  is  proportional  to  the  integrated  volume  velocity  over  the  surface  of  a 
plate. 


1247 


A  number  of  studies  have  been  carried  out  in  which  distributed  piezoelectric 
actuators  form  a  layer  of  a  laminated  system.  In  [7]  the  shape  of  a  distributed 
piezoelectric  actuator  was  chosen  to  be  orthogonal  to  all  but  one  of  the  natural 
modeshapes  of  the  cylindrical  shell  system  xmder  control.  Using  this 
approach  a  set  of  actuators  could  be  matched  to  the  modes  of  the  system 
under  control,  avoiding  control  spillover  {i.e.  the  excitation  of  tmcontrolled 
structural  modes). 

In  the  present  paper  a  single  shaped  PVDF  actuator  is  applied  to  a  thin  plate 
to  control  the  noise  transmission  through  it.  The  shape  of  the  actuator  is 
chosen  specifically  to  apply  an  approximately  uniform  force  to  the  plate. 
Such  an  actuator  can  be  used  to  cancel  the  total  volume  velocity  of  the  plate 
and  therefore  substantially  to  reduce  the  radiated  sound  power.  (If  volume 
velocity  is  measured  at  the  plate  surface  there  is  no  requirement  for  a  remote 
error  microphone.)  As  noted  by  Johnson  and  Elliott  [6],  the  soimd  power 
radiated  by  a  plate  which  is  small  compared  with  an  acoustic  wavelength 
depends  mainly  on  the  volume  velocity  of  the  plate.  The  simulations  in  [6] 
show  that  provided  the  plate  is  no  larger  than  half  an  acoustic  wavelength,  a 
single  actuator  used  to  cancel  volume  velocity  will  achieve  similar  results  to  a 
strategy  in  which  radiated  power  is  minimised.  It  is  possible  to  envisage  a 
large  partition  made  up  of  a  number  of  active  plate  elements  designed  on  this 
basis. 


CALCULATION  OF  NORMAL  FORCES  IN  THE  PLATE 

In  this  section  the  equation  of  motion  of  the  plate  and  attached  PVDF  layers  is 
set  out.  The  film  thickness  is  assumed  to  be  0.5  mm.  The  analysis  broadly 
follows  that  of  Dimitriadis,  Fuller  and  Rogers  [8],  but  the  individual 
piezoelectric  coefficients  and  included  separately  as  is  appropriate 

for  PVDF  and  a  sensitivity  function  is  included  to  account  for  variations  in 
electrode  shape.  In  addition  the  bending  stiffness  of  the  piezoelectric  film  is 
included  (it  is  not  negligible  as  the  whole  plate  is  covered).  The  nomenclature 
matches  that  used  by  Fuller,  Elliott  and  Nelson  [9]  but  here  the  analysis  leads 
to  the  inhomogeneous  wave  equation  for  the  plate-actuator  system. 

We  consider  an  aluminium  plate  of  thickness  Ih^  as  shown  in  Figure  1.  The 
plate  is  covered  on  its  upper  and  lower  surfaces  with  a  piezoelectric  film  of 
sensitivity  (}>(x,y)d-p  in  which  d.^  is  the  strain/ electric  field  matrix  of  the  material 
(3x6  array)  and  <^{x,y)  is  a  spatially-varying  sensitivity  function 
(0  <  (j)(x,y)  <  1 ).  The  two  piezoelectric  films  are  assumed  to  be  identical  but 
the  same  drive  voltage  is  applied  with  opposite  polarity  to  the  lower  film.  As 
a  result  of  this  antisymmetric  arrangement,  the  plate  is  subject  to  pure 
bending  with  no  straining  of  the  plate  midplane. 


1248 


In  line  with  other  similar  calculations  [2,9]  it  is  assumed  that  any  line 
perpendicular  to  the  midplane  before  deformation  will  remain  perpendicular 
to  it  when  the  plate /PVDF  assembly  is  deformed.  As  a  result,  the  strain  at 
any  point  in  the  assembly  is  proportional  to  distance  z  through  it.  (z=0  is 
defined  to  be  on  the  midplane  of  the  plate  as  indicated  in  Figure  1.)  The 
direct  and  shear  strains  throughout  the  whole  assembly  (e^,  Ey,  E^y)  are  then 
given  by  [10]: 


3^^ 


d^w 


(1-3) 


in  which  w  is  the  displacement  of  the  midplane  in  the  z  direction. 

The  corresponding  stresses  in  the  plate  (only)  follow  from  Hooke's  law  as  in 
the  standard  development  for  thin  plates: 


p  _ _ p_ 


= 


i-v: 


A  ^  p 

0''  = — — - 


2a+vj 


(4-6) 


in  which  is  the  Young's  modulus  and  Vp  is  Poisson's  ratio  for  the  plate 
material.  Stress  in  the  piezoelectric  film  follows  from  the  constitutive 
equations  for  the  material  [2].  The  direct  and  shear  stresses  for  the  upper 
piezo  film  are  designated  of  of  and  respectively: 


's/ 

K 

or"’ 

=  C  ^ 

1 - 

CO 

_^36. 

in  which  is  the  voltage  applied  across  the  actuator  fibn  (thickness  /rj  and 
4  and  are  the  strain/field  coefficients  for  the  material.  For  PVDF  d^,  = 
0,  but  it  is  included  in  the  analysis  for  completeness.  As  in  [2],  the  stiffness 
matrix  C  is  given  by: 


1249 


0 


c 


^pe  ^  pe^pe 

1  — 

^  ^  pe  ^  pe 

^  pe^pe  ^pe 

1-v'  1-v^ 

0  0 


0 

^pe 

2(l  +  v^J 


(8) 


The  stresses  in  the  lower  piezoelectric  layer  are  designated  and 

,  and  the  form  of  the  expression  is  similar  to  the  upper  layer  except  that 
the  voltage  is  applied  with  reversed  polarity: 


's/ 

^31 

V 

=  c 

+  (^{x,y) 

a 

Summing  moments  about  the  x  and  y  axes  for  a  small  element  dxby  of  the 
plate  yields  the  moment  per  unit  length  about  the  y-axis  and  My  about 
the  x-axis;  also  the  corresponding  twisting  moments  per  imit  length,  My^  and 


M^=jc!’’zdz+ 

-III,  ~K-K  ^‘b 

(10) 

h,,  -III,  ‘‘b+lh, 

My  =  j  a^yZdz  +  J  +  J  cf'^zdz 

-III,  -hh-K  h 

(11) 

h,,  -III,  l>h+lh, 

=  |<t;,z*+  ja^fhdz 

(12) 

-III,  -K-K  h 


and  My^=M,y. 


The  vertical  acceleration  at  each  point  of  the  plate  d^w/dt^  is  obtained  by 
taking  moments  about  the  x  and  y  axes  for  a  small  element  6x6y  and 
resolving  vertical  forces  as  in  standard  thin  plate  theory.  If  the  plate  is  acted 
on  by  some  external  force  per  unit  area  p(x,y)f(t)  then  the  vertical  motion  of 

the  plate  is  described  by: 


dx^  dxdy  dy^  dt^ 


=  -p(x,y)f(t) 


(13) 


1250 


in  which  m  is  the  mass/ area  of  the  plate-film  assembly. 


The  equation  of  motion  of  the  plate  complete  with  attached  piezoelectric  film 
is  obtained  by  combining  the  above  equations.  For  convenience  the  following 
constants  are  defined; 


n  _ 

''  3(1 


(bending  stiffness  of  plate,  thickness  2/i,)  (14) 


3(1 -V^) 


(bending  stiffness  due  to  PVDF)  (15) 


+2/1,) 


The  equation  of  motion  of  the  plate  assembly  including  upper  and  lower 
piezoelectric  layers  is  then  given  by: 


3^(l)(^,y) 


3>(j:,y) 


The  left-hand  side  of  this  equation  determines  the  free  response  of  the  plate- 
film  assembly  and  is  recognised  as  the  standard  form  for  a  thin  plate.  The 
first  term  on  the  right-hand  side  is  the  assumed  externally  applied  normal 
force  per  imit  area.  The  second  term  on  the  right-hand  side  gives  the  effective 
normal  force  per  unit  area  applied  to  the  plate  due  to  the  two  piezoelectric 
films  driven  by  a  voltage  .  It  is  clear  that  this  force  depends  on  the 
spatially-varying  sensitivity  which  has  been  assumed  for  the  piezoelectric 
material. 

An  examination  of  Eqns.  (16)  and  (17)  shows  that  the  normal  force  applied  by 
the  piezoelectric  film  depends  on  the  sum  of  the  plate  thickness  and  the 
thickness  of  one  of  the  film  layers.  If  the  film  is  much  thinner  than  the  plate 
(h^ «  hy)  then  the  applied  force  becomes  independent  of  the  film  thickness  and 
depends  only  on  the  plate  thickness,  the  applied  voltage,  the  electrode  pattern 
and  the  material  constants.  Eqn.  (17)  also  shows  that  the  normal  force  is 
applied  locally  at  all  points  on  the  plate.  No  integration  is  involved,  and  so  in 
contrast  with  a  volume  velocity  sensor  designed  using  quadratic  strips  [5],  the 
force  does  not  depend  on  the  plate  boundary  conditions  in  any  way. 
Furthermore,  there  is  no  need  to  use  two  films  oriented  at  90  degrees  to  cancel 

1251 


the  cross-sensitivity  It  is  also  worth  noting  that  no  assumption  has  been 
made  about  the  modeshapes  on  the  plate. 

We  can  create  a  uniform  force  actuator  by  choosing: 

^  =  constant,  and  (18) 

Bx 

(19) 

This  can  approximately  be  achieved  by  depositing  electrodes  in  the  form  of 
narrow  strips  whose  width  varies  quadratically  in  the  x-direction.  (Note  that 
the  x-direction  is  defined  as  the  direction  of  rolling  of  the  PVDF  material,  Le. 
the  direction  of  for  maximum  sensitivity.)  The  form  of  the  electrodes  is  as 
shown  in  Figure  2.  With  this  pattern  the  sensitivity  function  takes  the  form: 

(t)(x,>')  =  (20) 

where  is  the  length  of  the  strip.  Thus  (t)(-^>^)  -  ^  at  x  =  0  and  x  =  (no 
electrode),  while  (|)(x,  31)  =  1  halfway  along  at  x  =  LJ2  (electrode  fully  covers 
the  film). 


APPLICATION  TO  A  THIN  ALUMINIUM  PLATE 

If  a  plane  wave  of  sound  pressure  level  94  dB  (say)  is  normally  incident  on  a 
hard  surface,  it  will  exert  a  pressure  of  2  Pa  rms  on  that  surface.  If  this 
incident  pressure  is  counterbalanced  by  a  uniform  force  actuator  applied  to  a 
plate,  then  the  plate  could  in  principle  be  brought  to  rest.  Thus  for  active 
control  of  everyday  noise  levels  the  uniform  force  actuator  will  need  to  be 
able  to  generate  a  normal  force /area  of  a  few  pascal  over  the  surface  of  the 
plate.  (When  the  incident  wave  impinges  on  the  plate  at  an  oblique  angle, 
many  natural  modes  of  the  plate  will  be  excited  and  it  will  not  be  possible  to 
bring  it  perfectly  to  rest  with  a  single  actuator;  however  it  will  remain 
possible  to  cancel  the  plate  volume  velocity  as  explained  earlier.) 

By  way  of  example  an  aluminium  plate  of  thickness  1  mm  will  be  assumed, 
with  a  free  surface  measuring  300  x  400  mm.  Attached  to  each  side  is  a  PVDF 
film  of  thickness  0.5  mm.  One  electrode  of  each  panel  would  be  masked  to 
give  quadratic  strips  of  length  300  mm  as  shown  in  Figure  2.  (The  width  of  the 
strips  is  unimportant,  but  should  be  significantly  smaller  than  the  structural 
wavelength  of  modes  of  interest  on  the  plate.)  In  this  case  it  turns  out  that 

=  6.64  for  the  plate,  and 


1252 


Dp,  =  1.27  for  the  PVDF  film. 

The  piezoelectric  constants  for  the  film  are  typically 

4  =  23  X  m/V  and 

4  =  3  X  lO’"'  m/V,  giving 

Cp,  =  3.28x10^ 

Finally  the  double  derivative  of  the  sensitivity  function  turns  out  to  be 

=  88.9  (300  mm  strip  length) 

dx^ 

The  bending  stiffness  of  the  1  mm  aluminium  plate  is  increased  by  20%  due  to 
the  addition  of  two  layers  of  PVDF  film  of  thickness  0.5  mm  each.  The  force 
per  unit  area  due  to  the  actuator  is  obtained  from  Eqn.  (17): 

force/area  =  4,(^3, +Vp,4)|^'^3 

=  6.96x10’^  V3  Pa 

Thus  1000  volt  rms  would  yield  a  tmiform  force/area  of  close  to  7  Pa.  This  is 
not  an  impracticable  voltage  level,  but  previous  experience  at  ISVR  suggests 
that  care  would  need  to  be  taken  to  avoid  electrical  breakdown  through  the 
air  between  electrodes,  or  over  damp  surfaces. 


ACTIVE  CONTROL  OF  SOUND  TRANSMISSION 

In  reference  [6]  Johnson  and  Elliott  have  presented  simulations  of  the  active 
control  of  harmonic  sound  transmitted  through  a  plate  using  a  uniform  force 
actuator.  Their  actuator  might  be  realised  along  the  lines  described  in  this 
paper.  In  the  simulatioi\s  presented  in  [6]  the  uniform  force  actuator  is  used 
with  a  matched  volume  velocity  sensor  having  the  same  electrode  shape  [4]. 
The  advantage  of  this  configuration  is  that  the  actuator  can  be  used  to  drive 
the  net  volume  velocity  of  the  plate  to  zero  without  exciting  high  order 
structural  modes  in  the  process  (control  spillover).  Simulations  of  a 
300x380x1  mm  aluminium  plate  showed  that  reductions  in  transmitted  sound 
power  of  around  10  dB  were  achievable  in  principle  up  to  600  Hz  using  this 
matched  actuator-sensor  arrangement. 

A  further  advantage  of  the  distributed  matched  actuator-sensor  pair  is  that 
the  secondary  path  through  the  plate  (for  active  control)  is  minimum  phase 
[6],  giving  good  stability  characteristics  if  a  feedback  control  loop  is 
implemented  to  control  random  incident  soimd  for  which  no  reference  signal 
is  available. 


1253 


CONCLUSION 


A  design  of  distributed  piezoelectric  actuator  has  been  presented  which 
generates  a  roughly  uniform  force  over  the  surface  of  a  plate.  An  example 
calculation  shows  that  the  design  is  capable  of  controlling  realistic  soimd 
pressure  levels.  When  used  in  combination  with  a  matched  volume  velocity 
sensor,  the  actuator-sensor  pair  will  have  minimum-phase  characteristics  and 
will  offer  the  possiblity  of  feedback  control  in  which  neither  a  reference  signal 
nor  a  remote  error  sensor  will  be  required. 


ACKNOWLEDGEMENT 

The  financial  support  of  the  European  Community  under  the  Framework  IV 
programme  is  gratefully  acknowledged.  (Project  reference:  BRPR-CT96-0154) 


REFERENCES 

[1]  G.M.  Sessler  (1981)  JAcoust  Soc  Am  70(6)  Dec  1981 1596-1608 
Piezoelectricity  in  polyvinylidene  fluoride 

[2]  C.K.  Lee  (1990)  JAcoust  Soc  Am  87(3)  Mar  1990  1144-1158  Theory  of 
laminated  piezoelectric  plates  for  the  design  of  distributed  sensors/ actuators. 
Part  I:  Governing  equations  and  reciprocal  relationships 

[3]  R.L.  Clark  and  C.R.  Fuller  (1992)  JAcoust  Soc  Am  91(6)  June  1992  3321-3329 
Modal  sensing  of  efficient  acoustic  radiators  with  polyvinylidene  fluoride 
distributed  sensors  in  active  structural  acoustic  control  approaches 

[4]  M.E.  Johnson,  S.J.  Elliott  and  J.A.  Rex  (1993)  ISVK  Technical  Memorandum 
723.  Volume  Velocity  Sensors  for  Active  Control  of  Acoustic  Radiation 

[5]  M.E.  Johnson  and  S.J.  Elliott  (1995)  Proceedings  of  the  Conference  on  Smart 
Structures  and  Materials  27  Feb-3  Mar  1995,  San  Diego,  Calif.  SPIE  Vol  2443. 
Experiments  on  the  active  control  of  sound  radiation  using  a  volume  velocity 
sensor 

[6]  M.E.  Johnson  and  S.J.  Elliott  (1995)  JAcoust  Soc  Am  98(4)  Oct  1995  2174- 
2186.  Active  control  of  sound  radiation  using  volume  velocity  cancellation 

[7]  H.S.  Tzou,  J.P.  Zhong  and  J.J.  Hollkamp  (1994)  Journal  of  Sound  and 
Vibration  177(3)  363-378  Spatially  distributed  orthogonal  piezoelectric  shell 
actuators:  theory  and  applications 

[8]  E.K.  Dimitriadis,  C.R.  Fuller  and  C.A.  Rogers  (1991)  Transactions  of  the 
ASME,  Journal  of  Vibration  and  Acoustics  113  100-107  Piezoelectric  actuators  for 
distributed  vibration  excitation  of  thin  plates 

[9]  C.R.  Fuller,  S.J.  Elliott  and  P.A.  Nelson  (1996)  Active  Control  of  Vibration. 
Academic  Press,  London. 

[10]  G.B.  Warburton  (1976)  The  Dynamical  Behaviour  of  Structures,  2nd  Edition. 
Pergamon  Press,  Oxford. 


1254 


layers  of 
PVDF  film 


Figure  1 :  Schematic  diagram  of  thin  plate 
covered  on  both  sides  with  a  layer  of  PVDF  film 


Figure  2:  Electrode  pattern  of  quadratic  strips  for 
uniform-force  actuator 


1255 


1256 


CONTROL  OF  SOUND  RADIATION  FROM  A  FLUID-LOADED  PLATE 
USING  ACTIVE  CONSTRAINING  LAYER  DAMPING 

J.  Ro,  A.  Al-Ali  and  A.  Baz 

Mechanical  Engineering  Department 
The  Catholic  University  of  America 
Washington  D.  C.  20064 

Abstract 

Sound  radiation  from  a  vibrating  flat  plate,  with  one  side  subjected  to 
fluid-loading,  is  controlled  using  patches  of  Active  Constrained  Layer  Damping 
(ACLD).  The  fluid-structure-controller  interaction  is  modeled  using  the  finite 
element  method.  The  damping  characteristics  of  the  ACLD/plate/fluid  system  are 
determined  and  compared  with  the  damping  characteristics  of  plate/fluid  system 
controlled  with  conventional  Active  Control  (AC)  and/or  Passive  Constrained 
Layer  Damping  (PCLD)  treatments.  Such  comparisons  are  essential  in 
quantifying  the  individual  contribution  of  the  active  and  passive  damping 
components  to  the  overall  damping  characteristics,  when  each  operates  separately 
and  when  both  are  combined  to  interact  in  unison  as  in  the  ACLD  treatments. 

I.  INTRODUCTION 

When  a  structure  is  in  contact  with  or  immersed  in  a  fluid,  its  vibration 
energy  radiates  into  the  fluid  domain.  As  a  result,  there  is  an  observable  increase 
in  the  kinetic  energy  of  the  structure  due  to  the  fluid  loading.  Because  of  this 
kinetic  energy  increase,  the  natural  frequencies  of  structures  which  are  subjected 
to  fluid-loading  decrease  significantly  compared  to  the  natural  frequencies  of 
structures  in  vacuo.  Therefore,  through  understanding  of  the  interaction  between 
the  elastic  plate  structures  and  the  fluid  loading  has  been  essential  to  the  effective 
design  of  complex  structures  like  ships  and  submarine  hulls.  Lindholm  et  al.  [1] 
used  a  chordwise  hydrodynamic  strip  theory  approach  to  study  the  added  mass 
factor  for  cantilever  rectangular  plates  vibrating  in  still  water.  Fu  et  al  [2]  studied 
the  dry  and  wet  dynamic  characteristics  of  vertical  and  horizontal  cantilever 
square  plates  immersed  in  fluid  using  linear  hydroelasticity  theory.  Ettouney  et 
al  [3]  studied  the  dynamics  of  submerged  structures  using  expansion  vectors, 
called  wet  modes  which  are  finite  series  of  complex  eigenvectors  of  the  fluid- 
structure  system.  Recently  Kwak  [4]  presented  an  approximate  formula  to 
estimate  the  natural  frequencies  in  water  from  the  natural  frequencies  in  vacuo. 

When  the  structure  and  the  fluid  domains  become  rather  complex, 
solutions  of  fluid-structure  coupled  system  can  be  obtained  by  finite  element 


1257 


methods.  Marcus  [5],  Chowdhury  [6],  Muthuveerappan  et  al.  [7]  and  Rao  et  al 
[8]  have  successfully  implemented  the  finite  element  method  to  predict  the 
dynamic  characteristics  of  elastic  plates  in  water.  Everstine  [9]  used  both  finite 
and  boundary  element  methods  to  calculate  the  added  mass  matrices  of  fiilly- 
coupled  fluid-structure  systems. 

The  above  investigations  formed  the  bases  necessary  to  devising  passive 
and  active  means  for  controlling  the  vibration  of  as  well  as  the  sound  radiation 
from  fluid-loaded  plates.  Passive  Constrained  Layer  Damping  (PCLD)  treatments 
have  been  used  extensively  and  have  proven  to  be  effective  in  suppressing 
structural  vibration  as  reported,  for  example,  by  Jones  and  Salerno  [10],  Sandman 
[11]  and  Dubbelday  [12].  Recently,  Gu  and  Fuller  [13]  used  feed-forward  control 
algorithm  which  relied  in  its  operation  on  point  forces  to  actively  control  the 
sound  radiation  from  a  simply-supported  rectangular  fluid-loaded  plate. 

In  the  present  study,  the  new  class  of  Active  Constrained  Layer  Damping 
(ACLD)  treatment  is  utilized  as  a  viable  alternative  to  the  conventional  PCLD 
treatment  and  Active  Constrained  (AC)  with  PCLD  treatment  (AC/PCLD).  The 
ACLD  treatment  proposed  combines  the  attractive  attributes  of  both  active  and 
passive  damping  in  order  to  provide  high  energy  dissipation-to-weight 
characteristics  as  compared  to  the  PCLD  treatment.  Such  surface  treatment  has 
been  successfully  employed  to  control  the  vibration  of  various  structural  members 
as  reported,  for  example,  by  Shen  [14]  and  Baz  and  Ro  [15].  In  this  paper,  the  use 
of  the  ACLD  is  extended  to  the  control  of  sound  radiation  from  fluid-loaded 
plates.  Finite  element  modeling  of  the  dynamics  and  sound  radiation  of  fluid- 
loaded  plates  is  developed  and  validated  experimentally.  Particular  focus  is 
placed  on  demonstrating  the  effectiveness  of  the  ACLD  treatment  in  suppressing 
the  structural  vibration  and  attenuating  the  sound  radiation  as  compared  to 
conventional  PCLD  and  AC/PCLD. 

This  paper  is  organized  in  five  sections.  In  Section  1,  a  brief  introduction 
is  given.  In  Section  2.,  the  concepts  of  the  PCLD,  ACLD  and  AC/PCLD 
treatments  are  presented.  In  Section  3,  the  dynamical  and  fluid  finite  element 
models  are  developed  to  describe  the  interaction  between  the  plate,  ACLD  and  the 
contacting  fluid.  Experimental  validation  of  the  models  are  given  in  Section  4. 
Comparisons  between  the  theoretical  and  experimental  performance  are  also 
presented  in  Section  4  for  different  active  and  passive  damping  treatments. 
Section  5,  summarizes  the  conclusions  of  the  present  study. 

2.  CONCEPTS  OF  PCLD,  ACLD  AND  AC/PCLD  TREATMENTS 

Figures  (1-a),  (1-b)  and  (1-c)  show  schematic  drawings  of  the  PCLD, 
ACLD  and  AC/PCLD  treatments  respectively.  In  Figure  (1-a),  the  plate  is  treated 


1258 


Figure  (1)  -  Schematic  drawing  of  different  surface  treatments  (a)  PCLD,  (b) 
ACLD  and  (c)  AC/PCLD. 


with  a  viscoelastic  layer  which  is  bonded  directly  to  the  plate.  The  outer  surface 
of  the  viscoelastic  layer  is  constrained  by  an  inactive  piezo-electric  layer  in  order 
to  generate  shear  strain  y,  which  results  in  dissipation  of  the  vibrational  energy  of 
the  plate.  Activating  the  constraining  layer  electrically,  generates  a  control  force 
Fp  by  virtue  of  the  piezo-electric  effect  as  shown  in  Figure  (l-b)  for  the  ACLD 
treatment  .  Such  control  action  increases  the  shear  strain  to  yj  which  in  turn 
enhances  the  energy  dissipation  characteristics  of  the  treatment.  Also,  a  restoring 
moment  Mp=d2Fp  is  developed  which  attempts  bring  the  plate  back  to  its 
undeformed  position.  In  the  case  of  AC/PCLD  treatment,  shown  in  Figure  (l-c), 
two  piezo- films  are  used.  One  film  is  active  and  is  bonded  directly  to  the  plate  to 
control  its  vibration  by  generating  active  control  (AC)  force  Fp  and  moment 
Mp^djFp.  The  other  film  is  inactive  and  used  to  restrain  the  motion  of  the 


1259 


viscoelastic  layer  in  a  manner  similar  to  the  PCLD  treatment  of  Figure  (1-a).  In 
this  way,  the  AC  action  operates  separately  from  the  PCLD  action.  This  is  unlike 
the  ACLD  configuration  where  the  active  and  passive  control  actions  operate  in 
unison.  Note  that  in  the  ACLD  configuration,  larger  shear  strains  are  obtained 
hence  larger  energy  dissipation  is  achieved.  Furthermore,  larger  restoring 
moments  are  generated  in  the  ACLD  treatments  as  compared  to  the  AC/PCLD 
treatments  as  the  moment  arm  d2  in  the  former  case  is  larger  than  the  moment  arm 
d3  of  the  latter  case.  This  results  in  effective  damping  of  the  structural  vibrations 
and  consequently  effective  attenuation  of  sound  radiation  can  be  obtained. 


3.  FINITE  ELEMENT  MODELING 

3.1  Overview 

A  finite  element  model  is  presented  in  this  section,  to  describe  the 
behavior  of  fluid-loaded  thin  plates  which  are  treated  with  ACLD,  PCLD  and 
AC/PCLD  treatments. 

3.2  Finite  Element  Model  of  Treated  Plates 


(b)  (c) 

Figure  (2)  -  Schematic  drawing  of  plate  with  ACLD/AC/PCLD  patches. 


Figure  (2)  shows  a  schematic  drawing  of  the  ACLD  and  AC/PCLD 
treatments  of  the  sandwiched  plate  which  is  divided  into  N  finite  elements.  It  is 
assumed  that  the  shear  strains  in  the  piezo-electric  layers  and  in  the  base  plate  are 
negligible.  The  transverse  displacement  w  of  all  points  on  any  cross  section  of  the 
sandwiched  plate  are  considered  to  be  the  same.  The  damping  layers  are  assumed 
to  be  linearly  viscoelastic  with  their  constitutive  equations  described  by  the 
complex  shear  modulus  approach  such  that  G=G  (1+T|i).  In  addition,  the  bottom 
piezo-electric  layer  (AC)  and  the  base  plate  are  considered  to  be  perfectly  bonded 
together  and  so  are  the  viscoelastic  layer  and  the  top  piezo-electric  layer. 


The  treated  plate  elements  considered  are  two-dimensional  elements 
bounded  by  four  nodal  points.  Each  node  has  seven  degrees  of  fireedom  to 
describe  the  longitudinal  displacements  u,  and  v,  of  the  constraining  layer,  U3  and 
V3  of  the  base  plate,  the  transverse  displacement  w  and  the  slopes  w  and  W  y  of 
the  deflection  line.  The  deflection  vector  {5}  can  be  written  as: 

{5}  =  {u„v„U3,  V3,W,  W  ^W  y}"" 

=  [{n,}  {N3}  {Nj  {n.}  {n,}  {N,}^ 

where  {5"}  is  the  nodal  deflection  vector,  {Nj},  {Nj},  {N3},  {N4},  {N5},  {N5} ,,, 
and  {N5}  y  are  the  spatial  interpolating  vectors  corresponding  to  u„  v„  U3,  V3,  w, 
w^,  and  Wy  respectively.  Subscripts  ,x  and  ,y  denote  spatial  derivatives  with 
respect  to  x  and  y. 

Consider  the  following  energy  functional  ITp  for  the  treated  plate/fluid 

system: 

np  =  I(u-TK+w.-w^+wJdv,  (2) 

where  U  is  the  strain  energy,  T^  is  the  kinetic  energy,  is  work  done  by  external 
forces,  Wp  is  work  done  by  the  back  pressure  inside  the  fluid  domain,  is  work 
done  by  the  control  forces  and  moments  and  V  is  the  volume  of  the  plate.  These 
energies  are  expressed  as  follows 


I T,dv  =X  ^  1 1„  (*'  =  5{®T['^p]{®')  ’ 

|w,dV=j{5'f{F}, 


=  5'  Kj  5' 


“  V.l^dxdy 

„[Ui  hi  ‘J  “  ax^  Ui  ‘  hi  j  “  Sy^J 

where  i=l  for  ACLD  control  or  i=3  for  AC  control  (6) 


and  I  WpdV  ={5'}’"[n  ]{p'}. 


where  {p®}  is  the  nodal  pressure  vector  of  the  fluid  element.  In  the  above 
equation  []^],  [Mp],  {F},  [KJ  and  [Q]  are  the  plate  stiffness  matrix,  mass  matrix, 
external  forces  vector,  piezo-electric  forces  and  moments  matrix  and  plate/fluid 
coupling  matrix  as  given  in  the  appendix.  In  equation  (6),  d3i  32  are  the  piezo¬ 
strain  constants  in  directions  1  and  2  due  to  voltage  applied  in  direction  3.  The 
voltage  is  generated  by  feeding  back  the  derivative  of  the  displacement  5  at 

critical  nodes  such  that  j  where  is  the  derivative  feedback  gain 

matrix  and  C  is  the  measurement  matrix  defining  the  location  of  sensors. 

Minimizing  the  plate  energy  fimctional  using  classical  variational  methods 

such  that  |anp/a{6®}j  =  0  leads  to  the  following  finite  element  equation: 

{[K]-»lM.]){5'}-[n]{p'}  =  {F}  (8) 

where  co  is  the  frequency  and  [K]  =  [Kp]  +  [K^.]  is  overall  stiffriess  matrix. 

3.3  Finite  Element  Model  of  the  Fluid 

The  fluid  model  uses  solid  rectangular  tri-linear  elements  to  calculate  the 
sound  pressure  distribution  inside  the  fluid  domain  and  the  associated  structural 
coupling  effects.  The  fluid  domain  is  divided  into  fluid  elements.  Each  of 


1262 


these  elements  has  eight  nodes  with  one  degree  of  jfreedom  per  node.  The 
pressure  vector  is  expressed  by  p  =  [Nf]{p®}  and  [NJ  is  pressure  shape  function 
and  {p®}  is  nodal  pressure  vector. 

Considering  the  following  functional  Elf  of  fluid  domain  Craggs  [16] 

where  [Kf]  and  [MJ  are  the  fluid  stiffiiess  and  mass  matrices  as  given  in  the 
appendix.  Minimization  of  equation  (9)  such  that  {OTf/^lp^}}  =  0  yields  the 
fluid  dynamics  as  coupled  with  the  structural  vibration: 

([K,]-o)^[M,]){p'}-<a^[nf{5'}  =  {0}  (10) 

The  boundary  conditions  involved  are  of  the  form 

ap/an  =  0,  at  a  rigid  boundary 


a  p  /  a  n  =  -pp  9^5  /  9 , 
and  p  =  0. 


at  a  vibrating  boundary 
at  a  free  surface 


where  pf  is  fluid  density. 

3.4.  Solutions  of  the  Coupled  Plate/Fluid  System 

Combining  equations  (8)  and  (10)  gives 

■[K]-co^[m,]  -[fi]  Ip'l.pl  (11) 

[Kr]-CD^[M,]J  Lp'J  k 

At  low  frequencies,  the  fluid  pressure  is  in  phase  with  the  structural 
acceleration,  i.e.  the  fluid  appears  to  the  structure  like  an  added  mass.  However, 
as  the  frequency  increases  the  added  mass  effect  diminishes  and  the  damping 
effect,  i.e.  the  pressure  proportional  to  velocity,  increases.  For  an  incompressible 
fluid,  the  speed  of  sound  c  approaches  oo,  thus  the  mass  matrix  of  the  fluid  [MJ 
vanishes,  and  equation  (11)  can  be  simplified  to 

■[K]-cd=[m,]  -[fl]]  rs']  pi  (12) 

_  [KfiJkJ  k 

If  the  fluid-structure  coupled  system  has  free  boundary  surface,  then  [KJ  is 
non-singular  [Everstin,  1991]  and  the  nodal  pressure  vector  {p®}  can  be  eliminated 
from  equation  (12)  as  follows: 

{p'}=-o.^[K,nnr{5'}  (13) 


1263 


Hence,  equation  (12)  yields 


([K]-ffl^(K]+[Mj)){8')  =  {F)  (14) 

where  [MJ  is  added  virtual  mass  matrix  defined  by  [Muthuveerappan,  1979] 

[Mj  =  [a][K,]-'[£2f  (15) 

Equation  (14)  only  involves  the  unknown  nodal  deflection  vector  {5®}  of  the 
structure.  When  {F}=0,  equation  (14)  becomes  an  eigenvalue  problem,  the 
solution  of  which  yields  the  eigenvalues  and  eigenvectors.  The  nodal  pressures 
can  then  be  obtained  from  equation  (13)  when  the  nodal  displacements  are 
determined  for  any  particular  loading  on  plate. 

4.  PERFORMANCE  OF  PARTIALLY  TREATED  PLATES  WITH 
FLUID  LOADING 

In  this  section,  comparisons  are  presented  between  the  numerical 
predictions  and  experimental  results  of  the  natural  frequencies  and  damping  ratios 
of  a  fluid-loaded  plate  treated  with  ACLD,  PCLD  and  AC/PCLD.  The  effect  of 
the  Active  Control,  Passive  Constrained  Layer  Damping  and  Active  Constrained 
Layer  Damping  on  the  resonant  frequency,  damping  ratio,  attenuation  of  vibration 
amplitude  and  sound  radiation  are  investigated  experimentally.  The  vibration  and 
sound  radiation  attenuation  characteristics  of  the  fluid-structure  coupled  system 
are  determined  when  the  plate  is  excited  acoustically  with  broadband  frequency 
excitation  while  the  piezo-electric  layers  are  controlled  with  various  control  gains. 
The  experimental  results  are  compared  with  the  theoretical  predictions. 

4.1  Experimental  Set-up 

Figures  (3-a)  and  (3-b)  show  a  schematic  drawing  and  finite  element  mesh 
of  the  experimental  set-up  along  with  the  boundary  conditions  used  to  describe 
the  fluid-structure  system.  The  finite  element  mesh  includes:  24  plate-elements 
and  560  fluid-elements.  The  coupled  system  has  a  total  of  815  active  degree  of 
freedoms.  The  aluminum  base  plate  is  0.3m  long,  0.2m  wide  and  0.4inm  thick 
mounted  with  all  its  edges  in  a  clamped  arrangement  in  a  large  aluminum  base. 
The  aluminum  base  with  mounting  frame  sits  on  top  of  a  water  tank.  One  side  of 
the  base  plate  is  partially  treated  with  the  ACLD/AC/PCLD  and  the  other  side  is 
in  contact  with  water.  The  material  properties  and  thickness  of  piezo-electric 
material  and  the  viscoelastic  layer  listed  in  Table  (1).  The  size  of  the  combined 
piezo-electric  and  viscoelastic  patch  occupied  one-third  of  the  surface  area  of  the 
base  plate  and  it  is  placed  in  the  middle  of  plate  as  shown  in  Figure  (3-b).  A  laser 


1264 


sensor  is  used  to  measure  the  vibration  of  the  treated  plate  at  node  27  as  shown  in 
Figure  (3-b).  The  sensor  signal  is  sent  to  a  spectrum  analyzer  to  determine  the 
frequency  content  and  the  amplitude  of  vibration.  The  signal  is  also  sent  via 
analog  power  amplifiers  to  the  piezo-electric  layers  to  actively  control  the  sound 
radiation  and  structural  vibration.  The  radiated  sound  pressure  level  into  the  tank 
is  monitored  by  a  hydrophone  located  at  5.0  cm  below  the  plate  center.  This 
position  is  chosen  to  measure  the  plate  mode  (1,  1)  which  dominates  the  sound 
radiation.  The  hydrophone  signal  is  sent  also  to  the  spectrum  analyzer  to 
determine  its  frequency  content  and  the  associated  sound  pressure  levels. 


Figure  (3)  -  The  experimental  set-up,  (a)  schematic  drawing,  (b)  finite  element 
meshes. 


Table  (1)  -  Physical  and  geometrical  properties  of  the  ACLD  treatment 


Layer 

Thickness(m) 

Density  (Kg/m^) 

Modulus(MPa) 

viscoelastic 

S.OSxlO’" 

1104 

30** 

piezoelectric 

28xl0-‘ 

1780 

*  Young’s  modulus  **  Shear  modulus 


4.2  Experimental  Results 

Experimental  validation  of  the  dynamic  finite  element  model  of  the 
ACLD/plate  system  in  air  has  been  presented  by  Baz  and  Ro  [15]  in  detail.  Close 
agreement  was  obtained  between  the  theoretical  predictions  and  the  experimental 


1265 


measurements.  The  d)mamic  finite  element  model  is  therefore  valid  to  provide 
accurate  predictions. 

For  the  uncontrolled  treated  plate/fluid  system,  considered  in  this  study, 
the  experimental  results  indicate  that  coupling  the  plate  with  the  fluid  loading 
results  in  decreasing  the  first  mode  of  vibration  fi*om  59.475Hz  to  10.52Hz.  The 
coupled  finite  element  model  predicts  the  first  mode  of  vibration  to  decrease  firom 
57.91Hz  to  10.24Hz.  The  results  obtained  indicate  close  agreement  between  the 
theory  and  experiments. 

Figure  (4-a)  shows  a  plot  of  the  normalized  experimental  vibration 
amplitudes  for  the  fluid-loaded  plate  with  the  ACLD  treatment  using  different 
derivative  feedback  control  gains.  According  to  Figure  (4-a),  the  experimental 
results  obtained  by  using  the  ACLD  treatment  indicate  that  amplitude  attenuations 
of  1 1.36%,  48.25%  and  75.69%  are  obtained,  for  control  gains  of  2500,  5000,  and 
13500,  respectively.  The  reported  attenuations  are  normalized  with  respect  to  the 
amplitude  of  vibration  of  uncontrolled  plate,  i.e.  the  plate  with  PCLD  treatment. 
Figures  (4-b)  display  the  vibration  amplitudes  of  the  plate/fluid  system  with 
AC/PCLD  treatment  at  different  derivative  feedback  control  gains.  The 
corresponding  experimental  attenuations  of  the  vibration  amplitude  obtained  are 
4.6%,  20.29%,  54.04%  respectively. 


Figure  (4)  -  Effect  of  control  gain  on  normalized  amplitude  of  vibration  of  the 
treated  plate,  (a)  ACLD  control  and  (b)  AC/PCLD  control. 


Figures  (5-a)  and  (5-b)  show  the  associated  normalized  experimental 
sound  pressure  levels  (SPL)  using  ACLD  and  AC/PCLD  controllers,  respectively. 
The  normalized  experimental  SPL  attenuations  obtained  using  the  ACLD 
controller  are  26.29%,  50.8%  and  76.13%  compared  to  10.02%,  24.52%  and 
53.49%  with  the  AC/PCLD  controller  for  the  considered  control  gains.  Table  (2) 


1266 


lists  the  maximum  control  voltages  for  the  ACLD  and  AC/PCLD  controllers  for 
the  different  control  gains. 

It  is  clear  that  increasing  the  control  gain  has  resulted  in  improving  the 
attenuations  of  the  plate  vibration  and  the  sound  radiation  into  the  fluid  domain. 
It  is  evident  that  the  ACLD  treatment  has  produced  significant  vibration  and 
sound  pressure  level  attenuation  as  compared  to  the  attenuations  developed  by  the 
AC/PCLD  or  PCLD  treatments.  It  is  also  worth  emphasizing  that  the  ACLD 
treatment  requires  less  control  energy  than  the  conventional  AC/PCLD  treatments 
to  control  the  sound  radiation  from  the  plate. 


Figure  (5)  -  Effect  of  control  gain  on  normalized  sound  pressure  level  radiated 
from  the  treated  plate,  (a)  ACLD  control  and  (b)  AC/PCLD  control. 


Table  (2)  -  Maximum  control  voltage  for  the  ACLD/ AC/plate  system 


K. 

2500 

5000 

13500 

ACLD 

0 

21.75  V 

31.20V 

39.60V 

AC 

50.40V 

76.38V 

Figure  (6)  shows  the  mode  shapes  of  the  first  four  modes  of  the  treated 
plate  with  and  without  fluid-loading  as  obtained  experimentally  using 
STARMODAL  package.  Figure  (7)  shows  the  corresponding  theoretical 
predictions  of  the  first  four  mode  shapes.  Close  agreement  is  found  between 
experimental  measurement  and  theoretical  predictions. 

Figure  (8)  presents  comparisons  between  the  theoretical  and  experimental 
natural  frequencies  and  the  loss  factor  of  a  plate  treated  with  the  ACLD  and 
AC/PCLD  for  different  control  gains.  Close  agreement  between  theory  and 


1267 


experiment  is  evident.  Note  also  that  increasing  the  control  gain  has  resulted  in 
increasing  the  damping  ratio  for  both  ACLD  and  AC/PCLD  treatments.  The 
comparisons  emphasize  the  effectiveness  of  the  ACLD  treatment  in  acquiring  the 
large  damping  ratio  to  attenuate  the  structural  vibration  and  sound  radiation. 


Figure  (6)  -  Experimental  results  of  first  four  mode  shapes  of  treated  plate  (a) 
without  fluid  loading  and  (b)  with  fluid  loading. 


Figure  (7)  -  Theoretical  predictions  of  first  four  mode  shapes  of  treated  plate  (a) 
without  fluid  loading  and  (b)  with  fluid  loading. 


5.  SUMMARY 

This  paper  has  presented  theoretical  and  experimental  comparisons 
between  the  damping  characteristics  of  plates  treated  with  ACLD  and 


1268 


conventional  AC  with  PCLD  treatments.  The  dynamic  characteristics  of  the 
treated  plates  when  subjected  to  fluid  loading  is  determined  for  different 
derivative  control  gains.  The  fundamental  issues  governing  the  performance  of 
this  class  of  smart  structures  have  been  introduced  and  modeled  using  finite 
element  method.  The  accuracy  of  the  developed  finite  element  model  has  been 
validated  experimentally.  The  effectiveness  of  the  ACLD  treatment  in  attenuating 
structural  vibration  of  the  plates  as  well  as  the  sound  radiated  from  these  plates 
into  fluid  domain  has  also  been  clearly  demonstrated.  The  results  obtained 
indicate  that  the  ACLD  treatments  have  produced  significant  attenuation  of  the 
structural  vibration  and  sound  radiation  when  compared  to  PCLD  and  to  AC  with 
PCLD.  Such  favorable  characteristics  are  achieved  with  control  voltages  that  are 
much  lower  than  those  used  with  conventional  AC  systems.  The  developed 
theoretical  and  experimental  techniques  present  invaluable  tools  for  designing  and 
predicting  the  performance  of  the  plates  with  different  damping  treatments  and 
coupled  with  fluid  loading  that  can  be  used  in  many  engineering  applications. 


♦  PCLD  ■  ACLD,  K:d=l 3500  □  AC,  K:d=  13500 

•  ACLD,Kd=2500  O  AC,  Kd=2500 
A  ACLD,Kd=5000  A  AC,  Kd=5000 


Theoretical  Natural  Frequency  (Hz)  Theoretical  Damping  Ratio 

Figure  (8)  -  Comparison  between  theoretical  predictions  and  experimental  results, 
(a)  natural  frequency,  (b)  damping  ratio. 

ACKNOWLEDGMENTS 

This  work  is  funded  by  The  U.S.  Army  Research  Office  (Grant  number 
DAAH-04-93-G-0202).  Special  thanks  are  due  to  Dr.  Gary  Anderson,  the 
technical  monitor,  for  his  invaluable  technical  inputs. 


1269 


REFERENCES 

1.  Lindholm  U.  S.,  Kana,  D.  D.,  Chu,  W.  H.  and  Abramson,  H.  N.,  Elastic 
vibration  characteristics  of  cantilever  plates  in  water.  Journal  of  Ship 
Research,  1965, 9,  11-22. 

2.  Fu,  Y.  and  Price,  W.  G.,  Interactions  between  a  partially  or  totally  immersed 
vibrating  cantilever  plate  and  the  surrounding  fluid.  Journal  of  Sound  and 
Vibration,  1987, 118(3),  495-513. 

3.  Ettouney,  M.  M.,  Daddazio,  R.  P.  and  Dimaggio,  F.  L.,  Wet  modes  of 
submerged  structures  -  part  litheory.  Trans,  of  ASMS,  Journal  of  Vibration 
and  Acoustics,  1992, 114(4),  433-439. 

4.  Kwak,  M.  K.,  Hydroelastic  vibration  of  rectangular  plates.  Trans,  of  ASME 
Journal  of  Applied  mechanics,  1996,  63(1),  110-115. 

5.  Marcus,  M.  S.,  A  finite-element  method  applied  to  the  vibration  of 
submerged  plates.  Journal  of  Ship  Research,  1978,  22,  94-99. 

6.  Chowdury,  P.  C.,  Fluid  finite  elements  for  added  mass  calculations. 
International  Ship  Building  Progress,  1972, 19,  302-309. 

7.  Muthuveerappan  G.,  Ganesan,  N,  and  Veluswami,  M.  A.,  A  note  on  vibration 
of  a  cantilever  plate  immersed  in  water.  Journal  of  Sound  and  Vibration, 
1979,  63(3),  385-391. 

8.  Rao,  S.  N.  and  Ganesan,  N.,  Vibration  of  plates  immersed  in  hot  fluids. 
Computers  and  structures,  1985,  21(4),  111-1%! . 

9.  Everstine  G.  C.,  Prediction  of  low  frequency  vibrational  frequencies  of 
submerged  structures.  Trans,  of  ASME,  Journal  of  Vibration  and  Acoustics, 
1991, 113(2),  187-191. 

10.  Jones,  I.  W.  and  Salerno,  V.  L.,  The  vibration  of  an  internally  damped 
sandwich  plate  radiating  into  a  fluid  medium.  Trans,  of  ASME,  Journal  of 
Engineering  for  Industry,  1965,  379-384. 

11.  Sandman  B.  E.,  Motion  of  a  three-layered  elastic-viscoelastic  plate  under 
fluid  loading.  J.  of  Acoustical  Society  of  America,  1975,  57(5),  1097-1107. 

12.  Dubbelday,  P.  S.,  Constrained-layer  damping  analysis  for  flexural  waves  in 
infinite  fluid-loaded  plates.  Journal  of  Acoustical  Society  of  America,  1991, 
(3),  1475-1487. 

13.  Gu,  Y.  and  Fuller,  C.  R.,  Active  control  of  sound  radiation  from  a  fluid- 
loaded  rectangular  uniform  plate.  Journal  of  Acoustical  Society  of  America, 
1993,  93(1),  337-345. 

14.  Shen,  I.  Y.,  Bending  vibration  control  of  composite  plate  structures  through 
intelligent  constrained  layer  treatments.  Proc.  of  Smart  Structures  and 
Materials  Conference  on  Passive  Damping  ed.  C.  Johnson,  1994,  Vol.  2193, 
115-122,  Orlando,  FL. 


1270 


15.  Baz,  A.  and  Ro,  J.,  Vibration  control  of  plates  with  active  constrained  layer 
damping.  Journal  of  Smart  Materials  and  Structures,  1996,  5,  272-280. 

16.  Craggs,  A.,  The  transient  response  of  a  coupled  plate-acoustic  system  using 
plate  and  acoustic  finite  elements.  Journal  of  Sound  and  Vibration,  1971, 
15(4),  509-528. 

APPENDIX 

1.  Stiffness  Matrix  of  the  Treated  Plate  Element 

The  stiffoess  matrix  [Kp];  of  the  ith  element  of  the  plate/ACLD  system  is 
given  by  Baz  and  Ro  [15]: 

(A-1) 

where  [K,l  and  [KJi  denote  the  in-plane,  shear  and  bending  stifj&iesses  of 
the  ith  element.  These  stiffoess  matrices  are  given  by: 

j  =  layer  1,2, and 3  (A-2) 

and  [K.,1  =  I  jjB,]"[Dj,][B,]dxdy  j  =  layer  1, 2 and3  (A-4) 

with  G2  denoting  the  shear  modulus  of  the  viscoelastic  layer  and  the  matrices  [BJ, 


b1  =  :^ 


\({N2}-{N4)/d  +  {N,} 

(n,1  +fNj 


’  [Bb]  = 

2{N5},„_ 

{Naj 

[B.]  = 

{n. 

[®^p]  = 


(N.},.  +  {N3},.+h{Ns},„ 

{n.},,  +  +  {N2}^  +{n.)  „  +h{N4^, 


1271 


- 1 

0 

_ i 

1 - 

0 

Ei 

1  n 

and  fj,  1  EA 

1  n 

1-vJ 

Vj  1  u 

Vj  1  u 

I_ 

'  «  '“Vi 

0 

0 

i' 

_o  0  V. 

.  j=l,2and3  (A-5) 


where  h  =  (hi-h3)/2  and  d  =  (h2+hi/2+D)  with  D  denoting  the  distance  from  the 
mid-piane  of  the  plate  to  the  interface  with  the  viscoelastic  layer.  Also,  Ij 
represent  the  area  moment  of  inertia  of  the  jth  layer. 

2.  Mass  Matrix  of  the  Treated  Plate  Element 

The  mass  matrix  [Mp];  of  the  ith  element  of  the  plate/ACLD  system  is 
given  by: 

(A-6) 

where  [Mjp]j  and  [M^Jj  denote  the  mass  matrices  due  to  extension  and  bending  of 
the  ith  element.  These  matrices  are  given  by 

[Mi,].=p,h,££({N,}^{N,}+{N3}''{N,})dxdy  +  p,h,[  [({N3)'"{N3}  +  {NX{N4))dxdy 

+  ip2h2  £  {  +{N,}^{N,})dxdy 

and  [  ],  =  (p,h,  +  P3h3  +  P3h3)  £  £  [  N3  f  [  N3  ]dxdy  (A-7) 

where  {NJ  =  {N,}+{N3}+h{Ns},3  and  {N,}  =  {NJ+{N4}+h{N5},3 

3.  Control  Forces  and  Moments  Generated  by  the  Piezo-electric  Layer 

3.1  The  in-plane  piezo-electric  forces 

The  work  done  by  the  in-plane  piezo-electric  forces  {Fp}i  of  the  ith 
element  is  given  by: 

i{5'}-{Fp}rhi££%d>'dy  (A-8) 


where  j=l  for  ACLD  control  or  j=3  for  AC  control.  Also,  Ojp  and  Sjp  are  the  in¬ 
plane  stresses  and  strains  induced  in  the  piezo-electric  layers.  Equation  (A-8) 
reduces  to: 


1272 


='"•1 IKFK] 


for  k=l, 4 


3.2  The  piezo-electxic  moments 

The  work  done  by  the  piezo-electric  moments  {Mp}i  due  to  the  bending  of 
the  piezo-electric  layer  of  the  ith  element  is  given  by: 


5'  .MM 


=  hj  U,e,Ady 


(A-10) 


Where  Gjb  and  are  the  bending  stresses  and  strains  induced  in  the  piezo-electric 
layers.  Equation  (A-10)  reduces  to: 


Mpxk 

x.' 

Mpyk 

d32 

Mp,yk 

_  0 

fork=l,..,4 


(A-11) 


4.  Stiffness  and  Mass  Matrices  of  the  Fluid  Element 

The  stiffness  matrix  [KJj  and  mass  matrix  [MJ^  of  the  ith  element  of  the 
fluid  system  are  given  by: 


[B,rNdv 


(A-12) 

(A-13) 


[N,]>,]dv  (A-13) 

where  [b^ ]  =  [[N^],,  [N^]  ^  [N^]  and  c  is  the  sound  speed. 

5.  Coupling  matrix  of  the  Treated  Plate/Fluid  System 

The  coupling  matrix  [Q]  of  the  interface  element  of  the  structure/fluid 
system  can  be  presented  by: 

[q]=  f  [  {N5}''[Nf]dxdy  (A-14) 

Ja  Jb 


1273 


1274 


ANALYTICAL  METHODS  II 


DYNAMIC  RESPONSE  OF  SINGLE-LINK  FLEXIBLE 
MANIPULATORS 


E.  Manoach^  G.  de  Paz^  K.  Kostadinov^  and  F.  Montoya^ 

^  Bulgarian  Academy  of  Sciences,  Institute  of  Mechanics 
Acad.  G.  Bonchev  St.  Bl.  4;  1113  Sofia,  Bulgaria 
^  Universidad  de  Valladolid,  E.T.S.I.L  Dpto.  IMEIM. 

C/Paseo  del  Cauce,  s/n  47011-  Valladolid,  Spain 

1.  INTRODUCTION 

The  flexible-link  manipulators  have  many  advantages  over  the  traditional  stiff 
ones.  The  requirements  for  light-weight  and  energy  efficient  robotic  arms 
could  be  naturally  satisfied  by  using  flexible  manipulators.  On  the  other  hand 
the  application  of  the  robotic  arm  in  such  activities  as  positioning  in  electronic 
microscopes  and  disc-drivers,  hammering  a  nail  into  a  board  or  playing  tennis, 
also  forces  the  modeling  and  control  of  the  dynamic  behavior  of  flexible  link 
manipulators. 

In  most  cases  the  elastic  vibrations  which  arise  during  the  motion  must  be 
avoided  when  positioning  the  end  point  of  a  robotic  arm.  These  are  a  part  of 
the  reasons  that  cause  a  great  increase  of  the  publications  in  this  topic  in  recent 
years. 

In  most  papers  the  flexible  robotic  arms  are  modeled  as  thin  linear  elastic 
beams.  In  [1-3]  (and  many  others)  the  Bemouli-Euiler  beam  theory,  combined 
with  finite-element  technique  for  discretization  with  respect  to  the  space 
variables  is  used  for  modeling  and  control  of  single-link  flexible  manipulators. 
The  same  beam  theory,  combined  with  mode  superposition  technique  is  used 
in  [4].  Geometrically  nonlinear  beam  theories  are  used  in  [5,6]  for  the 
modeling  of  a  single-link  and  multi-link  flexible  robotic  manipulators, 
correspondingly. 

Taking  into  account  the  fact  that  robotic  arms  are  usually  not  very  thin  and  that 
the  transverse  shear  could  play  an  important  role  for  dynamically  loaded 
structures  [7]  the  application  of  the  Bernouli-Euiler  beam  theory  could  lead  to 
a  discrepancy  between  the  robotic  arm  behavior  and  that  one  described  by  the 
model. 


1275 


The  aim  of  this  work  is  to  model  the  dynamic  behavior  of  a  single  link  flexible 
robotic  arm  employing  the  Timoshenko  beam  theory,  which  considers  the 
transverse  shear  and  rotary  inertia.  The  arm  is  subjected  to  a  dynamic  loading. 
As  in  [3],  the  viscous  friction  is  included  into  the  model  and  slip-stick 
boundary  conditions  of  the  rotating  hub  are  introduced.  Besides  that,  the 
possibility  of  the  rise  of  a  contact  interaction  between  the  robotic  arm  and  the 
stop  (limiting  support)  is  included  into  the  model.  The  beam  stress  state  is 
checked  for  plastic  yielding  during  the  whole  process  of  deformation  and  the 
plastic  strains  (if  they  arise)  are  taken  into  account  in  the  model.  The 
numerical  results  are  provided  in  order  to  clarify  the  influence  of  the  different 
parameters  of  the  model  on  the  response  of  the  robotic  arm. 

2.  BASIC  EQUATIONS 
2.1.  Formulation  of  the  problem 

The  robotic  arm  -  flexible  beam  is  attached  to  a  rotor  that  has  friction  and 
inertia.  The  beam  is  considered  to  be  clamped  to  a  rotating  hub  and  its  motion 
consists  of  two  components:  “rigid-body”  component  and  a  component 
describing  the  elastic  deflection  of  the  beam  (see  Figure  1).  The  motion  of  the 
flexible  beam  is  accomplished  in  the  horizontal  plane  and  gravity  is  assumed 
to  be  negligible. 


Figure  1.  Model  of  one-link  flexible  manipulator. 


1276 


Tip  of  the  beam  (with  attached  tip-mass)  is  subjected  to  an  impulse  loading. 
Stick-slip  boundary  conditions  due  to  Coulomb  friction  of  the  hub  are 
introduced  when  describing  the  motion  of  the  beam.  In  other  words,  if  the 
bending  moment,  about  the  hub  axis,  due  to  the  impact  is  lower  than  the  static 
friction  torque  threshold  then  the  hub  is  considered  clamped  and  the  beam 
elastic  motion  is  considered  only.  When  the  bending  moment  exceeds  friction 
torque  threshold  this  boundary  condition  is  removed,  allowing  rotation  of  the 
hub  and  the  arm.  When  the  hub  speed  and  kinetic  energy  of  the  beam  become 
again  beneath  the  torque  threshold,  the  hub  clamps  again. 

The  possibility  of  the  rise  of  a  contact  interaction  between  the  robotic  and  the 
stop  is  envisaged.  In  this  case,  if  the  hub  angle  exceeds  the  limit  value,  the 
robotic  arm  clamps  and  a  part  of  the  beam  goes  in  a  contact  with  the  stop, 
which  is  modeled  as  an  elastic  foundation  of  a  Vinkler  type. 

In  view  of  the  fact  that  the  impact  loading  and  contact  interactions  are 
included  in  the  investigations,  it  is  expedient  to  be  considered  the  rise  of 
plastic  strains  in  the  beam. 


2.2.  Deriving  the  equations  of  motion. 


The  total  kinetic  and  potential  energy  of  the  rotating  hub  with  the  attached 
beam  (described  by  the  Timoshenko  beam  theory)  and  a  lumped  mass  at  it’s 
tip  can  be  expressed  as  follow: 

Ek  =^|pA[u(x,t)]  dx  +  |j„[e(t)]^+tMT[u(l,t)f jEl[(()(x,t)]  dx 


EIl^l  +kGA’ 
dx. 


V  dx 


dx 


(1  a,b) 


In  these  equation  u(x,t)  is  the  total  displacement 
u(x,t)  =  w(x,t)+x0(t)  (2) 

and  w  is  the  transverse  displacement  of  the  beam,  (p  is  the  angular  rotation, 
9  is  the  hub  angle,  E  is  the  Young  modulus,  G  is  the  shear  modulus.  A:  is  a 
shear  correction  factor,  p  is  the  material  density,  Jh  is  the  inertia  moment  of 
the  hub,  A=b*h  is  the  beam  cross-section  area,  h  is  the  thickness,  b  is  the 
width,  I=bh^/12 , 1  is  the  length  of  the  beam,  Mt  is  the  tip  mass,  t  is  the  time. 


1277 


Denoting  the  work  of  external  forces  (applied  actuating  torque  T(t)  and  the 
beam  loading  p(x,t))by 


W=T(t)e(t)+  \p{x,t)w{,x,t)dx 

0 

the  Hamilton's  principle  can  be  applied: 
j5(E,-Ep)dt+  j5Wdt  =  0 


(3) 

(4) 


Substituting  eqns  (l)-(3)  into  eqn  (4)  after  integrating  and  including  damping 
of  the  beam  material,  the  viscous  friction  of  the  hub  and  the  reaction  force  of 
the  elastic  foundation  the  following  equations  of  motion  can  be  obtained: 


.  r  .  X.  i2x  de  ,  ,5^w(l,t)  , 

(J  +  Jh  +  M-j-1  ^  j  ^2  ^  ^  ^  ^2 


i 

JpAx 


a^w(x,t) 

dt^ 


dx  =  T(t) 


EI^-^  +  kGA 
dx^ 


dw 


dx 


dw 


.d^cp 


- 9  “  <^2 - — 2  ~  0<x</ ,  t>0  (5a-c) 


dt 


dt 


kGA 


+  x- 


a^0 

'St" 


-R(x,t)  =  p(x,t) 


The  boundary  conditions  are: 
w(0,t)=(p(0,t)=0 


dcpgt)  Q 
dx 


kGA 


dw(l,t) 

dx 


d^w(lt) 

dt^ 


(6a-c) 


and  the  initial  conditions  are: 

wfx,  0)  =w^(x),  w(x,0)  =  w°  (x) ,  p(x, 0)=(f^(x),  (p(x,0)  =  9°  (x) 

e(t)  =  e(t)  =  o,  t<t,,p 

or  9(0  = 

when  |9(t)|  and  kinetic  energy  of  the  beam  falls  simultaneously  under  the 
Coulomb  friction  thresholds. 

In  eqns  (5)  the  viscous  friction  coefficient  of  the  hub  is  denoted  by  di,  d2  and 
d^  are  damping  coefficients  of  the  beam  material,  J  is  the  moment  of  inertia  of 
the  beam  about  the  motor  axis  and  R(x,t)  is  the  reaction  force  of  the  stop 
disposed  from  Xj  to  X2  (xj<X2<l)  and  modeled  as  an  elastic  foundation  with 
Vinkier  constant  r: 


(7a-d) 
(7  e,f) 
(7e)’ 


1278 


R(x,t)  = 


r[w(x,  t)  -  (x)]  for  x  j  <  x  <  x^ ;  |0l  > ' 

0  for  0<x<x,,X2  <x</;  iBi  < 


The  time  when  the  bending  moment  about  the  hub  axis  exceeds  the  friction 
torque  threshold  is  denoted  by  tsUp^  When  t>tsiip  this  condition  is  removed 
(allowing  rotation  of  the  hub  and  the  beam)  until  the  moment  when  the  beam 
clamps  again. 

2.3.  Elastic-plastic  relationships 

The  beam  stress-strain  state  is  usually  expressed  in  terms  of  generalized 
stresses  and  strains  which  are  function  of  x  coordinate  only.  As  a  unique 
yield  criterion  in  terms  of  moments  and  the  transverse  shear  force  does  not 
exist  according  to  Drucker  [8],  the  beam  cross-section  is  divided  into  layers 
and  for  each  of  them  the  stress  state  has  to  be  checked  for  yielding. 

The  relation  between  the  stress  vector  S  =  and  the  strain  vector 

s  =  |-z  ^ I  ’  generally  presented  as 


S=[D]8, 

where  in  the  case  of  an  elastic  material  [D]  =  [D^]  = 


is  the  elastic 


matrix  and  f(z)  is  a  function  describing  the  distribution  of  the  shear  strains 
along  the  thickness. 

On  the  basis  of  the  von  Mises  yield  criterion,  the  yield  surface  is  expressed 


After  yielding  during  infinitesimal  increment  of  the  stresses,  the  changes  of 
strains  are  assumed  to  be  divisible  into  elastic  and  plastic  parts 


As  =  As^  +  As^ 


where 


As 


1279 


By  using  eqns  (10),  (11)  and  the  associated  flow  rule  [9],  following  Yamada 
and  others  [10],  the  following  explicit  relationship  between  the  increments  of 
stresses  and  strains  is  obtained 


.  AS  =  [d‘'’]A8 

where  [-0'^]  is  the  elastic-plastic  matrix: 


HY\d^] 

LjaLiVi 

r^r 

1  J  1  J  L  j\asj 

1 - 

C/2 

1— 

108] 

In  this  equation  H  is  a  function  of  the  hardening  parameter.  For  ideal  plasticity 
H  is  equal  to  zero,  while  for  a  wholly  elastic  material  H  ->  co  . 

3.  Use  of  mode  superposition. 

3.1.  Rearrangement  of  the  equation  of  motion. 

Let  the  total  time  interval  T  on  which  the  dynamic  behavior  of  the  structures  is 
investigated,  be  divided  into  sequence  of  time  increments  y+J. 

In  the  numerical  calculations  the  following  dimensionless  variables  are  used: 
x  =  xl  U  w  =  w//,  t  =tl  / c  c  =  /  p 

and  then  omitting  the  bars,  and  after  some  algebra,  the  governing  equations 
can  be  written  in  the  following  form: 


d'e  de 

- T-  +  C,  - 

d  t^  '  d  t 


--cJ  w(l,t)-  f(p(x,t)dx  =C3  T(t)+  fxp(x,t)dx 
tv  0  /Vo  / 


09  0^9  -I  c/w  I 


(15  a-c) 


0^W  01V  0^iv  09 

dt^  ^  dt  1 0x  ^  dx 


=  -p~G{-G^ 


where  a=12P/h^  ,  p~kG/E,  p=pI/(EA),  c,=  d^l/(cJ}{),  Arp/(JHp)  C3= 
P/(c^J}^  ),  C4~d49/EI,  C3=djl/EA,  =5/6.  The  nonlinear  force  due  to  the 

reaction  of  the  foundation  is  denoted  by  G^  =  R(x,t).l/EA  and  Gf  and  G2  are 
the  components  of  the  so-called  non-linear  force  vector  Gp(  .G^}  which  is 
due  to  the  inelastic  strains.  It  has  the  presentation  (see  [7] ): 


1280 


3.2  Mode  superposition  method 

The  l.h.s.  of  eqns  (15  b,c)  is  a  linear  form  and  therefore  the  mode 
superposition  method  can  be  used  for  its  solution.  As  the  eigen  frequencies 
and  the  normal  modes  of  vibrations  of  an  elastic  beam  do  not  correspond  to  the 
real  nonlinear  system,  these  modes  are  called  "pseudo-normal"  modes. 

Thus,  the  generalized  displacements  vector  v  =  |a”^  (p,  w|  is  expanded  as  a 
sum  of  the  product  of  the  vectors  of  pseudo-normal  modes  v^  and  the  time 
dependent  functions  q„(t)  as 

(17) 

n 

The  analytically  obtained  eigen  functions  of  the  elastic  Timoshenko  beam  are 
chosen  as  basis  functions  (see  APPENDIX).  When  the  tip  mass  is  attached  to 
the  beam  the  eigen  functions  of  the  system  used  in  the  mode  superposition 
method  are  preliminary  orthogonalized  by  standard  orthogonalization 
procedure  as  it  is  mentioned  in  the  APPENDIX. 

Substituting  eqn  (17)  into  (15  b,c),  multiplying  by  v^(x),  integrating  the 
product  over  the  beam  length,  invoking  the  orthogonality  condition  (see  (A9) ) 
and  assuming  a  proportional  damping  for  the  beam  material 

|(c4(pf,  +c^wl)rdr  the  following  system  of  ordinary  differential 

equations  (ODE)  for  0(t)  and  qn(t)  is  obtained: 


^(f)  +  c,  (9(0  =  Cj  2]  [«'» (1)  -  ]?» 


^  » 


+  C3(r(0  +  P(0) 


(18a, b) 


9„(o+2f„ffl„9„(o+®,k,(o=-‘y„  -'f„&(o+&!’ 

In  this  equations 

1 

11  V  f 

Y„  =  J(p„  6„  =  =  jxw,Xx)ck,  P(t)=  \xp{x,t)  dx , 


1  1 

g;(t)=  |G'’(x,t)v„(x)dx,  g',(t)=  jG'(x.t)w„(x)dx,  co„  are  the  eigen 

0  0 

frequencies  of  the  elastic  clamped  beam  and  are  the  modal  damping 
parameters. 

The  initial  conditions  defined  by  eqns  (7  a-d)  are  transformed  also  in  terms  of 
q„(0),  and  ?„(0) 

9„(0)  =  9„“,  4„(0)  =  4°, 

q°„  =  J(w°w„  t-a'VVjdx,  (j“„  =  j(w"w„ +a-'(p°(p„)dx,  (19a-d) 

0  0 

The  obtained  system  of  nonlinear  ordinary  differential  equations  is  a  stiff  one 
and  it  is  solved  numerically  by  the  backward  differential  formula  method,  also 
called  the  Gear's  method  [11]. 

The  rise  of  plastic  strains  is  taking  into  account  by  using  an  iterative  procedure 
based  on  the  "initial  stresses"  numerical  approach  [7]. 


4.  RESULTS  AND  DISCUSSION 

Numerical  results  were  performed  for  the  robotic-arm  with  the  same  material 
and  geometrical  characteristic  as  these  given  in  [3]  in  order  to  make  some 
comparisons.  Model  parameters  are:  E=6.5xl0  Pa  ,v=0.2,  Cp=2.6xl0  Pa, 
1=0.7652,  b=0.00642  m,  h=0.016  m,  p=2590  kgW  ,  Mt=0,153  kg,  Jh=0.285 
kgm^  (Jh  is  not  defined  in  [3].) 

The  aim  of  the  computations  is  to  show  and  clarify  the  influence  of  the  elastic 
or  elastic-plastic  deformation  on  the  motion  of  the  robotic-arm,  to  demonstrate 
the  effect  of  the  hub  friction,  slip-stick  boundary  conditions  and  the  contact 
interaction  between  the  beam  and  the  stop. 


1282 


Only  impact  loading  on  the  beam  is  considered  in  this  work,  i.e.  dynamic  load 
p(x,t)  and  applied  torque  T(t)  are  equal  to  zero.  The  impact  loading  is 
expressed  as  an  initial  velocity  applied  to  the  tip  of  the  beam  0.95  <  x  <  1 . 

Nine  modes  are  used  in  expansion  (15)  but  the  results  obtained  with  number  of 
modes  greater  than  nine  are  practically  indistinguishable  from  these  shown 
here. 

For  all  calculations  the  material  damping  is  equal  to  8%  of  the  critical 
damping. 

The  results  for  the  rotation  of  the  hub  of  the  flexible  manipulator  with  an 
attached  mass  at  its  tip  -1  and  without  an  attached  mass  -2  are  shown  in  Figure 
2.  The  hub  friction  is  not  considered.  The  beam  is  subjected  to  an  impact 
loading  with  initial  velocity  =-1.95  m/s.  As  can  be  expected,  the  hub  angle 
increases  much  faster  in  the  case  of  the  beam  with  an  attached  mass.  The 
corresponding  beam  deflections  are  presented  in  Figure  3.  The  results  obtained 
are  very  close  to  these  obtained  in  [3]  (Fig.  7  and  8  in  [3]).  The  frequencies  of 
forced  vibrations  obtained  in  [3],  however,  correspond  to  the  beam  without  an 
attached  mass. 


0,  rad 


-4.00  I - ^ ^ ^ ^ - 1 - ^ ^ i ^ 

0,00  l.OO  2.00  3.00  4.00  5.00 

t,  sec 

Figure  2.  Hub  response  without 
viscous  friction.  1  -  beam  with  an 
attached  mass;  2-  without  an  attached 
mass 


w,  m 


Figure  3.  Deflection  of  the  tip  of  the 
beam  without  viscous  friction  of  the 
hub.  1  -  beam  with  an  attached  mass; 
2  -  without  an  attached  mass 


1283 


The  influence  of  the  hub  friction  on  the  flexible  manipulator  response  can  be 
seen  in  Figure  4.  The  viscous  friction  is  set  di=0.1  Nms,  the  static  Coulomb 
friction  threshold  is  equal  to  0.06  Nm  and  three  cases  of  the  hub  slip-stick 

threshold  are  tested:  |e|  stick  =  0-005  rad.s'^  -  1,  |0|  stick  =  0.0085  rad.s'^  , 

stick  =  0.01  rad.s'^  For  this  initial  velocity  (w°  =-1.95m/s)  the  hub  slips 

very  fast  from  the  initial  clamped  state  and  the  beam  begins  to  rotate.  As  can 
be  expected,  the  consideration  of  the  viscous  friction  of  the  hub  leads  to  a 
decrease  of  the  angle  of  rotation  of  the  beam  and  changes  the  linear  variation 
of  0  with  time.  The  results  show  also  that  the  value  of  the  hub  speed 

threshold  |0|  stick  exercises  an  essential  influence  on  the  motion  of  the 
rotating  system. 

When  |e|  =0.005  rad.s'  the 

beam  sticks  at  t  =  2.602  s  after  that 
the  hub  periodically  slips  and  sticks 
which  also  leads  to  damping  of  the 

motion.  When  joj  stick  “  0.0085 

rad.s'^  the  start  of  sticking  occurs  at 
t  =  1.7207  s  and  after  t=2.417  s  the 
hub  clamps  with  short  interruptions 
till  t=3.4  s  when  due  to  the  elastic 
vibrations  it  snaps  in  the  direction 
opposite  to  w°»  clamps  again  at 
3.679s,  slip  at  4.5s,  and  finally 

clamps  at  t=4.5s.  When  joj  stick 
=0.0085  rad.s"'  the  sticking  begins 
at  t= 1.0525s  and  very  fast  (at  t«2  s) 
the  beam  clamps  with  0=-O.587  rad. 

In  order  to  observe  the  occurrence  of  the  plastic  deformation  the  beam  was 
subjected  to  impulse  loading  having  larger  values  of  initial  velocities.  In 
addition,  the  contact  interaction  between  the  beam  and  the  stop  was 
considered. 

The  beam-tip  deflection  in  the  presence  of  a  contact  with  the  stop  disposed  at 
X  €[0.16,  0.263]  and  initial  velocity  w°  =-15.95  m/s  is  shown  in  Fig.  5. 


0,rad 


di=0.1Nms.  1-  |0|  stick”0-005rad.s"', 

2  -  |0|  stick  =  0-0085  rad  s"'  ; 

3-  |0|  stick  =0-0 is-' 


1284 


In  order  to  reduce  the  computational  time  the  limit  value  of  0  was  chosen 
e^j^p=0.0025  rad.  When  this  value  was  reached  the  problem  was  automatically 
resolved  with  new  initial  conditions  0=0,  w°=w(Xstop,tstop),  etc. 

As  can  be  seen,  the  presence  of  the  contact  interaction  during  the  process  of 
motion  of  the  beam  due  to  the  elastic  support  for  x  e[0.16,  0.263]  leads  to  a 
decrease  of  the  amplitudes  of  vibrations  in  the  direction  of  the  stop  in 
comparison  with  the  amplitudes  in  the  opposite  direction.  The  variation  of  the 
beam  displacements  along  the  beam  length  for  the  first  0.8  s  of  motion  is 
shown  in  Fig.  6.  It  must  be  noted  that  in  this  case  of  loading  a  plastic  yielding 
occurs.  It  is  assumed  that  beam  material  is  characterized  by  an  isotropic  linear 
strain  hardening  and  H=0.5.  The  plastic  yielding  occurs  simultaneously  with 
the  contact  interaction  at  t=0.01366  s  at  the  clamped  end  of  the  beam.  At 
t=0.0186  s  the  plastic  zone  spreads  to  x  =  0.158  and  at  t=0.08767  if  covers  the 
length  to  X  =0.3 1 .  The  last  points  that  yields  are  x  =0.55,  0.61  at  t=l .  1 19  s. 
Seven  layers  along  the  beam 
thickness,  symmetrically  disposed 
about  the  beam  axis  was  checked  for 
yielding  (N2=7)  but  the  plastic  zone 
has  reached  the  second  and  6th 
layers  only  at  the  clamped  end  of  the 
beam  (x  =0).  In  all  other  point  along 
beam  length  the  plastic  yielding 
occurs  only  at  the  upper  and  lower 
surface  of  the  beam.  The  plastic 
strains  are  small  and  the  response  of 
the  beam  is  not  very  different  from 
the  wholly  elastic  response. 

Nevertheless,  the  appearance  of  such 
kind  of  plastic  deformations  in  the 
structures  used  for  the  precise 
operations  must  be  taken  into 
account  in  the  manipulator  self 
calibration  procedure. 


w,m 

0.40 

0.30 

0.20 

0,10 

0.00 

-O.IO 

-0.20 


-0.30 


0.00 


1.00 


2.00 


3.00 


4.00  5.00 

t,  sec. 

Figure  5.  Deflection  of  the  tip  of  the 
beam  with  time  in  the  case  of  a 
contact  with  the  stop.  r=5.5xl0^  Pa 


5.  CONCLUSIONS 

In  this  work  a  model  describing  dynamic  behavior  of  a  deformable  beam 
attached  to  a  rotating  hub  that  has  friction  and  inertia  is  developed.  The 
Timoshenko  beam  theory  is  used  to  model  the  elastic  deformation  of  the  beam. 


1285 


Figure  6,  Variation  of  the  elastic-plastic  beam  displacement  along  the  beam 
length  with  time  in  the  case  of  a  contact  with  the  stop. 

The  slip-stick  boundary  conditions  are  also  incorporated  into  the  model 
The  possibility  of  the  rise  of  undesired  plastic  deformations  in  the  case  of  a 
high  velocity  impact  on  the  clamped  robotic  arm,  or  in  the  case  of  a  contact 
with  limiting  support  (stop)  is  included  into  the  model. 

The  analytically  obtained  eigen  functions  of  the  elastic  Timoshenko  beam 
vibrations  are  used  to  transform  the  partial  differential  equations  into  a  set  of 
ODE  by  using  the  mode  superposition  method.  This  approach  minimizes  the 
number  of  ODE  which  have  to  be  solved  in  comparison  with  another 
numerical  discretization  techniques  (finite  elements  or  finite  difference 
methods). 

The  results  obtained  show  the  essential  influence  of  elasticity  on  the  robotic- 
arm  motion. 

The  model  will  be  used  to  synthesise  a  control  of  one  link  flexible 
manipulators  and  for  a  self  calibration  procedure  when  plastic  deformation 
would  occur. 

Acknowledgments 

The  authors  gratefully  acknowledge  the  financial  support  from  EC  Copernicus 
Program  under  the  Project  ROQUAL  CIPA  CT  94  0109. 

The  first  author  wishes  to  thank  the  National  Research  Fund  for  the  partial 
financial  support  on  this  study  through  Contract  MM-5 17/95. 


1286 


REFERENCES 


1.  Bayo,  E.  A  finite-element  approach  to  control  the  end-point  motion  of  a 

single  -link  flexible  Robert.  J.  Robotic  System,  1987,  4„  63-75 

2.  Bayo,  E.  and  Moulin,  H.,  An  efficient  computation  of  the  inverse  dynamics 

of  flexible  manipulators  in  the  time  domain.  IEEE  Proc  Int.  Conf.  on 
Robotics  and  Automations,  1989,  710-15. 

3.  Chapnik,  B.V.,  Heppler,  G.R.,  and  Aplevich,  J.D.  Modeling  impact  on  a 

one-link  flexible  robotic  arm.  IEEE  Transaction  on  Robotics  and 
Automation,  1991,7,479-88. 

4.  Liu,  L.  and  Hac,  A.,  Optimal  control  of  a  single  link  flexible  manipulator. 

Advances  in  Robotics,  Mechatronics,  and  Haptic  Interfaces,  1993,  DCS- 
49,  303-13. 

5.  Wen,  J.T.,  Repko,  M.  and  Buche,  R.,  Modeling  and  control  of  a  rotating 

flexible  beam  on  a  translatable  base.  Dynamics  of  Flexible  Multibody 
Systems:  Theory  and  Experiment,  1992,  DCS-37,  39-45. 

6.  Sharan,  A.M.  and  Karla,  P.,  Dynamic  Response  of  robotic  manipulators 
using  modal  analysis.  Meek  Mach.  Theory,  1994,  29,  1233-49. 

7.  Manoach,  E.  and  Karagiozova,  D.  Dynamic  response  of  thick  elastic-plastic 

beams.  International  Journal  of  Mechanical  Sciences,  1993,  35,  909-19 

8.  Drucker,  D.C.  Effect  of  shear  on  plastic  bending  of  beams.  J.  of  Applied 

Mechanics,  1956, 23,  515-21 

9.  Hill,  R.  Mathematical  Theory  of  Plasticity,  1950,  Oxford  University  Press, 
London. 

10.  Yamada,  Y.,  Yoshimura,  N.  and  Sakurai  T.,  Plastic  stress-strain  matrix  and 
its  application  for  the  solution  of  elastic-plastic  problems  by  the  finite 
elements,  Int.  J.  of  Mechanical  Sciences  ,  1968, 10,  343-54 

11.  Gear,  C.W.,  Numerical  initial  value  problem  in  ordinary  differential 
equations,  1971,  Prentice-Hall,  Englewood  Cliffs,  NJ. 

12.  Abramovich,  H.  Elishakoff,  1.  Influence  of  shear  deformation  and  rotary 
inertia  on  vibration  frequencies  via  Love’s  equations.  J.  Sound  Vibr.,  1990, 
137,  516-22. 


1287 


APPENDIX:  NORMAL  MODES  OF  FREE  VIBRATIONS  OF  A 
CLAMPED  TIMOSHENKO  BEAM  WITH  AN  ATTACHED  MASS, 


Equations  (5  b,c)  can  be  decoupled,  transforming  them  into  two  fourth  order 
equations  [12]  as  regards  cp  and  w. 

Solving  this  equations  (with  p=0  and  R=0)  and  using  the  boundary  conditions 
(6)  (with  0=0) ,  the  equations  of  the  frequencies  and  forms  of  vibrations  of  the 
beam  are  obtained. 

Introducing  following  denotations 

1/2  (A1  a-e) 

^2„={®,^(l  +  P)  +  [®:a  +  P)^+4(B,^(a-pco,^,)]''^}  /2, 

/l„  =(4+CoJP)/5,„,  /2„  /3„  =(4-®SP)''-S!». 

the  frequencies  of  free  vibrations  are  determined  as  roots  of  equation: 


a)  In  the  case  >0  i.e.  co  <  a  /  p  the  frequencies  equation  is: 

^11^22  ”^12^21  “  ^  >  (-^2) 
where 

^11  ~ /2n‘^2n  ^12  ~  ■^/2«‘^2«  ®^^(‘^2«) 

h\  =  -f\nM.Sxn )  “  /2„  sin(52„ )  +  s^„sh(si, )  +  52„  sin(52„ )  +  ,  .  . 

(A3  a-d) 

M‘^lnCh(^lJ  +  ‘^2«  cos(,y2J] 


*22  =  /2j=OS(^2«)-cll('SlJ]  +  4^S,„ch(i,„)-52„  COS(52„)  + 

J\n 

+  sl„  sin(i2„)] 

Jin 


and  the  modes  of  vibrations  are: 

w„W  =  -S„ 

1 

'  b 


^(ch(i,„x)  -  cos(j2„x))  +  sin(i2„x)  -  ^sh(i|„x) 

J\» 


L^i 


(A4  a,b) 

(/,„sh(,s,„x)  +  sin(,y2«^))  +  fi,,  icos{s^„x)  -  ch(s,„x) 


b)  In  the  case  5^,,  <0  i.e.  >  a  /  P  the  eigen  frequency  equation  (A2)  has 
the  following  presentation: 


1288 


b\\  =  +  /2„i2„  C0S(J2„) 

bn  =  ?i„/2nSin(?i„)  -  /2„i2n  sin(j2„) 

621  =  /3„sin(ii„)  -  /2„  sin(52„)  -?i„sin(Ji„)  +  S2„  sinC^jJ  - 
>.[J,^„COS(  j,„  )  -  sl„  C0S(i2„)] 

*22  =  /2„[cOS(S2„)  -  COS(J,„)]  -  ^  S,„COS(?,„)  +  S2„  COS{s-^„)  - 

J\n 

-  sl„  sin(52„)] 

7  In 

sh  =  -5?„ 


and  the  of  vibrations  are: 

w„(x)  = 


f  b 

sin(j2„2c)  -  ^sin(?,„x)  -  ^(cosCij,,*)  -  cos(j,„x)) 
/i„  *11 


'P„W  =  -S» 


(A5  a-e) 


(A6a-b) 


/j,,  (cos(i2„x)  -  cos(^|„x))  -  ^(/3„sin(i,„x)  -  f^„  sin(jj„x)) 

0,, 


When  a  mass  is  not  attached  at  the  beam  tip  the  following  orthogonality 
condition  is  fulfilled: 

1 


p  fO,  n^  m; 

(a  _ 

J  [1,  n  -  m. 


(A7) 


and  when  an  attached  mass  is  considered  the  modes  are  orthogonalized  by 
standard  orthogonalization  procedure. 

The  constants  are  obtained  from  condition  (A7). 


1289 


1290 


Wave  Reflection  and  Transmission 
in  an  Axially  Strained,  Rotating  Timoshenko  Shaft 


B.  Kang  ‘  and  C.  A.  Tan  ^ 
Department  of  Mechanical  Engineering 
Wayne  State  University 
Detroit,  Michigan  48202,  U.  S.  A. 


Abstract 

In  this  paper,  the  wave  reflection  and  transmission  characteristics  of  an  axially  strained, 
rotating  Timoshenko  shaft  under  general  support  and  boundary  conditions,  and  with  geometric 
discontinuities  are  examined.  The  static  axial  deformation  due  to  an  axial  force  is  also  included 
in  the  model.  The  reflection  and  transmission  matrices  for  incident  waves  upon  these  point 
supports  and  discontinuities  are  derived.  These  matrices  are  combined,  with  the  aid  of  the 
transfer  matrix  method,  to  provide  a  concise  and  systematic  approach  for  the  free  vibration 
analysis  of  multi-span  rotating  shafts  with  general  boundary  conditions.  Results  on  the  wave 
reflection  and  transmission  coefficients  are  presented  for  both  the  Timoshenko  and  the  simple 
Euler-Bemoulli  models  to  investigate  the  effects  of  the  axial  strain,  shaft  rotation  speed,  shear 
and  rotary  inertia. 


‘  Graduate  Research  Assistant.  Tel:  +1-313-577-6823,  Fax:  +1-313-577-8789.  E-mail:  kang@feedback.eng.wayne.edu 
’  Associate  professor  (Corresponding  Author),Tel:  +1-313-577-3888,  Fax:+1-313.577-8789.  E-mail:  tan@tan.eng.wayne.edu 

Submitted  to:  Sixth  International  Conference  on  Recent  Advances  in  Structural  Dynamics,  Institute  of  Sound  and  Vibration, 
Southampton,  England,  July,  1997 


1291 


A. 

NOMENCLATURE 

Area  of  shaft  cross  section  [m^] 

do 

Diameter  of  shaft  cross  section  [m] 

C 

Generalized  coordinate  of  an  incident  wave  [m] 

Cdt  {Ct) 

Translational  damping  coefficient  [N-sec/m] 

Cdr  (Cr) 

Rotational  damping  coefficient  [N-m-sec/rad] 

Co 

Bar  velocity  [m/sec] 

Cj 

Shear  velocity  [m/sec] 

D 

Generalized  coordinate  of  a  transmitted  wave  [m] 

E,G 

Young’s  and  shear  modulus  [NW],  respectively 

I 

Lateral  moment  of  inertia  of  shaft  [m'^] 

Jm  (Jm) 

mass  moment  of  inertia  of  a  rotor  mass  [kg-m"^] 

K 

Timoshenko  shear  coefficient 

Knikr) 

Rotational  spring  [N/rad] 

Kt  (kd 

Translational  spring  [N/m] 

t 

Length  of  shaft  [m] 

M{m) 

Mass  of  rotor  [kg] 

P 

Axial  force  [N] 

rij ,  ti] 

Reflection  and  transmission  coefficients,  respectively,  i  -  1  positive  traveling 

U{u) 

wave;  i  =  2  negative  traveling  wave;  jr  =  1  propagating  wave  for  Cases  11  and 
/V;  7  =  2  attenuating  wave  for  Cases  II  and  IV.  Both  7  =1 ,  2  for  propagating 
wave  for  Case  I 

Transverse  displacement  [m] 

X-Y-Zix-y-z) 

Reference  frame  coordinates  [m] 

a 

{K-G)IE 

Rotation  parameter,  see  Eqn.  (Id) 

£ 

Pl{E-A),  axial  strain 

e' 

Non-dimensional  axial  load  parameter,  see  Eqn.  (13b) 

T,  f  (r,  y) 

Wavenumber  [m‘‘] 

nAi) 

See  Eqns.  (20a,  b),  (22a,  b)  and  (24a,  b) 

p 

Mass  density  of  shaft  [kgW] 

G 

Diameter  ratio  between  two  shaft  elements 

57,  (ft)) 

System  natural  frequency  for  Timoshenko  model  [rad/sec] 

a 

System  natural  frequency  for  Euler-Bernoulli  model  [rad/sec] 

Q. 

Rotation  speed  of  shaft  [rad/sec] 

W  (¥) 

Bending  angle  of  the  shaft  cross-section  [rad] 

subscript  L,  r 

The  left  and  right  side  of  a  discontinuity,  respectively. 

superscript  -,+ 

Negative  and  positive  traveling  waves,  respectively,  when  used  in  C  and  D. 

Note:  Symbols  in 

Otherwise  denotes  quantities  on  the  left  and  right  side  of  a  discontinuity, 
respectively 

parenthesis  are  the  corresponding  non-dimensional  parameters. 

1292 

1.  INTRODUCTION 


The  vibrations  of  elastic  structures  such  as  strings,  beams,  and  plates  can  be  described  in 
terms  of  waves  propagating  and  attenuating  in  waveguides.  Although  the  subject  of  wave 
motions  has  been  considered  much  more  extensively  in  the  field  of  acoustics  in  fluids  and  solids 
than  mechanical  vibrations  of  elastic  structures,  wave  analysis  techniques  have  been  employed  to 
reveal  important,  physical  characteristics  associated  with  vibrations  of  structures.  One  advantage 
of  the  wave  technique  is  its  compact  and  systematic  approach  to  analyze  complex  structures  such 
as  trusses,  aircraft  panels  with  periodic  supports,  and  beams  on  multiple  supports  [1].  Previous 
works  based  on  wave  propagation  techniques  have  been  well  documented  in  several  books  [2-4], 
Recently,  Mead  [5]  applied  the  phase-closure  principle  to  determine  the  natural  frequencies  of 
Euler-Bemoulli  beam  models.  A  systematic  approach  including  both  the  propagating  and  near- 
field  waves  was  employed  to  study  the  free  vibrations  of  Euler-Bemoulli  beams  [6]. 

High  speed  rotating  shafts  are  commonly  employed  in  precision  manufacturing  and  power 
transmission.  Despite  the  usefulness  of  the  wave  propagation  method  in  structural  vibrations, 
applications  of  this  technique  to  study  the  dynamics  and  vibrations  of  a  flexible  shaft  rotating 
about  its  longitudinal  axis  have  seldom  been  considered.  The  purpose  of  this  paper  is  to  examine 
the  wave  reflection  and  transmission  [6]  in  an  axially  strained,  rotating  Timoshenko  shaft  under 
various  support  and  boundary  conditions.  The  effect  of  the  axial  load  is  included  by  considering 
the  axial  static  deformations  in  the  equations  of  motion.  This  paper  is  a  sequel  to  another  paper 
in  which  the  authors  discuss  the  basic  wave  motions  in  the  infinitely  long  shaft  model  [7]. 

Although  there  have  been  numerous  studies  on  the  dynamics  and  vibration  of  rotating  shafts, 
none  has  examined  the  effects  of  axial  strains  (which  cannot  be  neglected  in  many  applications) 
on  the  vibration  characteristics  of  a  Timoshenko  shaft  under  multiple  supports.  Modal  analysis 
technique  has  been  applied  to  study  the  vibration  of  a  rotating  Timoshenko  shaft  with  general 
boundary  conditions  [8,  9],  and  subject  to  a  moving  load  [10].  Recently,  the  distributed  transfer 
function  method  was  applied  to  a  rotating  shaft  system  with  multiple,  geometric  discontinuities 
[11],  The  wave  propagation  in  a  rotating  Timoshenko  shaft  was  considered  in  Ref.  [12].  Other 
major  works  on  the  dynamics  of  rotating  shafts  have  been  well  documented  in  Refs.  [13-15]. 

This  manuscript  is  organized  as  follows.  Governing  equations  of  motion  [16]  and  basic  wave 
solutions  for  the  Timoshenko  shaft  are  outlined  in  Section  2.  Each  wave  solution  consists  of  four 
wave  components:  positive  and  negative,  propagating  and  attenuating  waves.  In  Section  3,  the 
wave  reflection  and  transmission  matrices  are  derived  for  the  shaft  under  various  point  supports 
and  boundary  conditions.  The  supports  may  include  translational  and  rotational  springs  and 
dampers,  and  rotor  mass.  Results  are  presented  for  both  the  Timoshenko  and  the  simple  Euler- 
Bemoulli  models  to  assess  the  effects  of  axial  strain,  shaft  rotation,  shear  and  rotary  inertia.  The 
wave  propagation  across  a  shaft  with  geometric  discontinuities  such  as  a  change  in  the  cross- 
section  is  examined  in  Section  4,  and  the  wave  reflection  at  a  boundary  with  arbitrary  support 
conditions  is  considered  in  Section  5. 

With  the  wave  reflection  and  transmission  matrices  as  the  main  analytical  tools,  it  is  shown  in 
Section  6  how  to  apply  the  current  results  together  with  the  transfer  matrix  method  to  analyze  the 


1293 


free  vibration  of  a  rotating,  multi-span  Timoshenko  shaft  system  in  a  systematic  manner.  The 
proposed  approach  is  then  demonstrated  by  considering  the  free  vibration  of  a  two-span  beam 
with  an  intermediate  support. 


2.  FOKMULATION  AND  WAVE  SOLUTIONS 


Consider  a  rotating  shaft  subjected  to  axial  loads  and  with  multiple  intermediate  supports  and 
arbitrary  boundary  conditions,  as  shown  in  Fig.  1.  Including  the  effects  of  rotary  inertia,  shear 
deformations,  and  axial  deformations  due  to  the  axial  loads,  the  uncoupled  equations  of  motion 
governing  the  transverse  displacement  u  and  the  slope  \}/  due  to  bending  can  be  derived  in  the 
following  non-dimensional  form 


..  .  d'^u  „  d^u 

£ 

+16a(l  +  £)(l-i-e - )^r^  =  0  , 

a  dr 


U  lA  £  .  d~U 

■2ri3^  +  a— -16£{1  +  £--)^^ 


a  dz'^ 


-l-16a(l-l-£)(l  +  £-— =  0  , 
a  dr 


u  =  —  z  =  —  r  =  —  T  =  ^  ■ 
a.'  a.'  T.  ^  \  KG 


Figure  1.  A  rotating  Timoshenko  shaft  model  subject  to  axial  loads  and  with  general  boundary  conditions. 


1294 


(Id) 


a 


KG 
E  ’ 


E 


Note  that  u  and  y/are  the  measurements  in  the  complex  plane,  that  is  u=ux+iuy  and  \i/=\}fx+iYr 
E  denotes  the  Young’s  modulus,  p  the  mass  density,  As  the  area  of  the  cross  section,  ao  the 
diameter  of  shaft,  K  the  Timoshenko  shear  coefficient,  G  the  shear  modulus  and  Q  the  constant 
angular  velocity  of  the  shaft.  Details  of  deriving  these  equations  of  motion  are  found  in  Ref. 
[16]. 

Assuming  and  substituting  the  following  wave  solutions  into  Eqns.  (la)  and  (lb) 

=  (2a) 

=  (2b) 

and  defining  the  non-dimensionalized  wavenumber  7  and  system  natural  frequency  w  gives  the 
frequency  equation,  Eqn.  (3a);  see  Ref.  [7], 


y  =  r^o  . 


_  CO  a, 


KG  . 


(j)  = - a.  =  - is  known  as  the  shear  velocity) . 


where, 


7“*  -A7"  4-5  =  0  , 


A  =  (1  -i-  a)a)  ^  -  2j3  cu  “  16£  (1  +  e - ) , 


5  =  a  ot)^  -  2j3  u)  -  16a  (1 4- £)(1 4- e  -  •^) 


The  four  roots  of  Eqn.  (3a)  are 


7  =  ±-^[a  ±  -x/a^  -45  . 


(2c) 

(2d) 

(3a) 

(3b) 

(3c) 

(4) 


In  general,  7  is  complex.  Let  (o  be  real.  It  can  be  shown  that,  with  a  >  0  and  £  the  axial  strain 

of  the  elastic  solid,  the  discriminant  A^  -45  is  positive  semi-definite  for  most  engineering 
applications.  Hence,  it  is  possible  to  classify  the  wave  solutions  into  four  distinct  cases.  Note 
that  one  may  study  the  wave  propagation  by  considering  only  a  single  general  form  of  the  wave 
solution.  However,  the  classification  procedure  identifies  the  coupled  modes  of  vibration  of  the 


1295 


Timoshenko  shaft  model  and  provides  a  better  understanding  on  how  each  wave  solution  governs 
the  wave  motions  [7].  Based  on  the  algebraic  relationships  between  A  and  B,  the  four  valid  wave 


solutions  are  obtained  as  follows. 

Case  /  ( A  >  0  and  B  >  0 ); 

«(z,  t)  =  +  C- (5a) 

V(z,  t)  =  (CJ.e-'*’"  +  )e®  (5b) 

Case  !I{A>0  andB<0): 

«(z,  t)  =  (Ce-"'''  +  +  C:^e-^'-‘  +  (6a) 

yz(z,t)  =  (c;,^-'^"  -  c;,/"  +  +  0;^/=-  )£®  (6b) 

CaseIIIiA<0mdB>0): 

uiz,t)  =  (C>-'''  +  +  C:,«")«'“  (7a) 

Vr(z,0  =  (C;,a-"  +  C;,ef>'  +  +  C;je'=')«®  (7b) 

Coje/V(A<0  andB<0): 

k(zA)  =  (C,>-^'^  +C,>’‘'-  +  C>-'''=‘  +  C,V'’'*')«'®  (8a) 

V/(z,«)  =  (C;,e-f-  +  (8b) 

where, 

f,  =^(|A|  +  V-5’-4|B|f,  r.  (9^.  W 

r,  =  +  4|B|  +  |A|)^ ,  r,  +  4|B|  -  |A|j' ,  (9c,d) 


and  the  coefficients  C*  and  C"  denote  positive-  and  negative-travelling  waves  from  the  origin  of 
disturbance,  respectively.  Important  remarks  on  the  basic  wave  propagation  characteristics  are 
summarized  from  [7].  First,  the  wave  solution  of  Case  III  does  not  exist  in  the  real  frequency 
space  since  this  type  of  solution  represents  a  situation  in  which  none  of  the  wave  components  can 
propagate  along  the  waveguide.  Therefore  the  study  of  Case  III  is  excluded  in  the  present  paper. 
Second,  the  vibrating  motion  of  the  shaft  model  in  Case  I  is  predominately  pure  shear  [17]  which 


1296 


is  unique  for  the  Timoshenko  shaft  model,  while  in  Case  U  and  Case  IV  the  flexural  mode  and 
the  simple  shearing  mode,  which  are  corrected  by  including  the  rotary  inertia  and  shearing  effects 
in  the  formulation,  dominate.  Third,  when  the  shaft  rotates  at  a  very  high  speed  and/or  the  shaft 
is  axially  strained  by  tensile  loads,  the  wave  solution  of  Case  IV  governs  the  vibrating  motion  of 
the  shaft  model  in  the  low  frequency  range. 


For  comparison,  the  parameters  A  and  B  in  the  simple  Euler-Bernoulli  beam  model  are 


(10a) 

(10b) 

where,  the  non-dimensionalized  natural  frequency  6)  is  defined  as 


is  known  as  the  bar  velocity). 


(10c) 


Note  that,  because  B  is  negative,  wave  solutions  of  Case  I  and  Case  III  do  not  exist. 

In  general  the  displacement  and  the  rotation  of  an  infinitesimal  shaft  element  consist  of  four 
wave  components  as  shown  by  Eqns.  (5a-8b).  Once  the  displacement  and  the  bending  slope  are 
known,  the  moment  M  and  shear  force  V  at  a  cross  section  can  be  determined  from 


M  =  EI^, 
dz 

(ll) 

(du  ^ 

y  = J . 

(12) 

Moreover,  the  kinematic  relationship  between  the  transverse  displacement  and  the  slope  due  to 
bending  is 

S^u  3^u  .  ,3  r 

3r~  3z^  3z  ' 

(I3a) 

where  e'  denotes  the  effects  of  the  axial  force  and  is  defined  as 

e  =  l  +  e  — . 

a 

(I3b) 

1297 


3.  WAVE  REFLECTION  AND  TRANSMISSION  AT  SUPPORTS 


When  a  wave  is  incident  upon  a  discontinuity,  it  is  transmitted  and  reflected  at  different  rates 
depending  on  the  properties  of  the  discontinuity.  Consider  a  rotating  Timoshenko  shaft  model 
supported  at  ^  =  0 ;  see  Fig.  2.  The  support  simulates  a  bearing  modelled  by  linear,  translational 
and  rotational  springs,  dampers,  and  a  rotor  mass  which  typically  represents  a  gear  transmitting  a 
torque.  Based  on  Eqns.  (5a-8b),  group  the  four  wave  components  into  2x1  vectors  of  positive¬ 
travelling  waves  and  negative-travelling  waves  C" ,  i.  e.. 


Recall  that,  depending  on  the  system  parameters,  the  rotating  Timoshenko  shaft  model  has  four 
(practically  three)  different  wave  solutions  in  the  entire  frequency  region  as  described  in  Eqns. 
(5a-8b).  Thus  C^  and  C,  in  the  above  expression  do  not  always  correspond  to  propagating  and 

attenuating  wave  components,  respectively.  When  a  set  of  positive-travelling  waves  is 
incident  upon  the  support,  it  gives  rise  to  a  set  of  reflected  waves  C~  and  transmitted  waves  . 
These  waves  are  related  by 


C'=rC" 

(15) 

D"  =  tC", 

(16) 

where  r  and  t  are  the  2x2  reflection  and  transmission  matrices  respectively  and  are  expressed  as 


r  = 

'^u 

^12* 

(17) 

/2! 

''22. 

’^11 

^12 

t  = 

(18) 

/2I 

^22  _ 

From  Eqns.  (5a-8b),  suppressing  term  and  excluding  Case  III,  the  displacements  u~  and 
and  the  bending  slopes  y/~and  y/'^  at  the  left  and  right  of  z  =  0,  respectively,  can  be  expressed 
in  terms  of  the  wave  amplitudes  of  the  displacement.  For  convenience,  the  over-bar  (•)  on  the 
wavenumbers  is  dropped  hereafter. 


Case  1  {A  >0  andB>0); 

«-(z)  =  C, +  c;,£'^==,  (19a) 

W'U)  =  -n,C:,e-‘^“  -77, c;, +T7,C>-'>''- (19b) 


1298 


M.Jm 


Figure  2.  Wave  motion  at  a  general  support  (the  disk  may  be  considered  as  a  gear  transmitting  a  torque). 


where, 

_  72-®' 

T/i  —  /  »  ~  /  • 

r,£  72^ 

//  ( A  >  0  and  B  <  0 ): 

«-(z)  =  c;, +  C'e"'''  +  , 

i^-(z)  =  7,, C, >-'"■=  -77,C,-^r''''=  +)7,C*e-''“  -TJ,C,;;e"=S 


k*(z)  =  A>“’'"  +  AV’‘. 


where 


= 


r,£' 


>  Ba  “ 


CaselV{A<0a.ndB<0): 


(19c) 

(19d) 

(20a,  b) 

(21a) 

(21b) 

(21c) 

(21d) 

(22a,  b) 


1299 


the  following  set  of  matrix  equations  can  be  established  for  each  Case. 
Case  /  ( A  >  0  and  5  >  0 ): 


n  n 

r  1 

1  1 

.  ri  n 

c^  + 

rC^=: 

L^i  12] 

-^2j 

-im,  Ic+r  IrC" 

KYi-tIi)  i(r2“7?2)J  [-KYi-Vi)  -i(.Y2-Tl2)\ 


(28a) 


ri,(k,-J„,co^)  +  iTi,{c,0)~r,)  0)")  + 1772 72 )  ^28b) 

ik^~mO)-)  +  i{c,co  +  r^-r]^)  (fc, -mtt>")  +  z’(c,®  +  72 -772)  J 


1300 


Case  //  ( A  >  0  and  S  <  0 ): 


1 

.^1 


1 


1 


L^i 


1 

^2 


tc^ 


1 

1 

n 

-ir,T7, 

[i(r, -T7i)  r2-j772_ 

C  -r 

-^r, -77,) 

~  (r,  -  it], )_ 

(29a) 


r},(k^-J„^0)-)  +  irij  (c^co-r^) 
(k,  -mco^)  +  i(c,Q)  +  r,-T],) 


Tl2(k,-J„,co^-r2)  +  iV2CrO} 

{kj  -m(0^  +r2)+/(c,Ct}-  772) 


tc", 


(29b) 


Case  /y(A<0and5<0); 


'1  11 

■  1  1  ■ 

rC"  = 

'  1  r 

Jli  Hi. 

-^2  -^1. 

jii  ^1- 

■-1X2772  -£t,77,  ■ 

C'* 

-1X2772 

KT.-ni)  r, -i77,_ 

-r 

—  i(r2  —  772 ) 

-(r,  -  in,). 

(30a) 


T]2ik^-J„^(0'")  +  iT]^{c,C0-y2) 
{k,  -m(D‘)  +  i(c,aj  +  r2  -772) 


riiik,-J„co^-y,)  +  iT],c^co' 
{k,  - mco^  +  r, )  +  i{c,(0 -rij) 


tC", 


(30b) 


where  Eqns.  (15)  and  (16)  have  been  applied  in  all  Cases.  Note  that  in  Eqn.  (27a),  it  is  assumed 
that  the  rotational  spring  at  the  support  is  attached  to  the  cross  section  of  a  shaft  element  such 
that  the  rotational  spring  responds  only  to  the  slope  change  due  to  rotation  of  the  cross  section 
and  not  the  total  slope  change  of  the  neutral  axis  of  the  shaft  model.  This  assumption  allows  the 
shearing  motion  of  the  shaft  element  at  the  support.  Note  also  that  the  effect  of  axial  loads  on  the 
shear  force  at  the  support  is  neglected  since  the  contribution  of  axial  loads  to  the  shear  force  at 
the  support  or  boundary  is  small  compared  to  the  shear  force  due  to  the  flexural  motion  of  the 
shaft  element.  Exact  moment  and  force  balance  conditions  at  boundaries  for  a  rotating 
Timoshenko  shaft  element  subjected  to  axial  loads  can  be  found  in  Ref.  [16]. 

The  corresponding  matrix  equations  for  the  simple  Euler-Bernoulli  shaft  model  are  shown  in 
Appendix  I.  Solving  the  set  of  matrix  equations  simultaneously  for  r  and  t  gives  the  elements  of 
the  reflection  and  transmission  matrices  for  each  Case.  The  general  forms  of  solutions  to  these 
sets  of  equations  for  each  Case  is  not  presented  in  this  paper  due  to  space  limitation.  However 
one  can  obtain  the  solutions  in  either  closed-form  or  numerically.  Note  that  in  Case  II  and  Case 
IV,  the  first  columns  of  r  and  t  are  the  reflection  and  transmission  coefficients  due  to  incident 
propagating  wave  components,  and  the  second  columns  are  due  to  an  incident  attenuating  wave 
component  which  is  generally  termed  as  near-field  since  this  type  of  wave  decays  exponentially 
with  distance.  When  the  distance  between  the  origin  of  disturbance  and  the  discontinuity  is  very 


1301 


large,  these  attenuating  wave  components  can  be  neglected.  However,  as  mentioned  by  many 
authors,  for  example  Graff  [2],  attenuating  waves  play  an  important  role  in  wave  motions  by 
contributing  a  significant  amount  of  energy  to  the  propagating  wave  components  when  a  set  of 
propagating  and  attenuating  waves  are  incident  at  a  discontinuity  and,  in  particular,  when  the 
distances  between  the  discontinuities  are  relatively  small,  as  in  the  case  of  closely-spaced  multi¬ 
span  beams.  In  this  paper,  near-field  components  are  included.  In  what  follows,  the  effects  of 
the  point  supports  on  the  reflection  and  transmission  of  an  incident  wave  are  studied.  For 
comparison,  the  results  are  obtained  for  both  the  Timoshenko  and  the  simple  Euler-Bemoulli 
models,  which  hereafter,  for  brevity,  are  denoted  by  TM  and  EB,  respectively.  The  system 
parameters  used  in  the  numerical  results  are  taken  from  Ref.  [10];  ao  -  0.0955  m,  p  =  7700 
kgW,  K  =  0.9,  E  =  207x10^  Wnf,  G  =  77.7x10^  N/ml 


3.1.  Wave  reflection  and  transmission  at  rigid  supports 

Consider  two  cases:  the  simple  support  and  the  clamped  support.  The  r  and  t  are  solved  and 
shown  as  follows. 

•  Simple  support  (k,  =  oo,  =  m=  c,  =  -  7„,  =0) 

Case  7  ( A  >  0  and  B  >  0 ): 


r  = 


_ 1 _ 

(72 -7i)(r  1/2+6;') 


7i(6>"-72) 
72(7? -6;-) 


7i(6;’-72) 


(31a) 


t  = 


_ 1 _ 

(72-7i)(7i72+6)-) 


72(7?-®') 


Case  77  ( A  >  0  and  S  <  0): 

_ 1 _ 


_ 1 _ 

(iT,-r,xr,r,-!V) 


Case  7V(A<0andB<0): 


_ 1 _ 


7i(6)"“72) 
7i(72 -6)') 


r,(r,H®-)  1 


r,{r^  +  co^)' 


(31b) 


(32a) 


(32b) 


(33a) 


1302 


The  corresponding  reflection  and  transmission  matrices  for  the  EB  model  are  listed  in  App.  I 

Figures  3  and  4  plot  the  moduli  (magnitudes)  of  the  reflection  and  transmission  coefficients 
for  the  simple  and  clamped  supports.  The  finite  cutoff  frequencies,  above  which  all  waves 
propagate,  are  also  marked  in  the  figures.  Thus,  for  the  TM  model,  the  wave  motions  change 
from  Case  11  to  Case  1  when  (0>C0^  {0)^-4-  in  Fig.  3,  co,  =  4.24  in  Fig.  4;  O),  is  slightly  altered 
by  rotation  speed  and  axial  load).  The  results  show  that,  at  low  frequencies  (cD  <  0.1  =  3156 
rad/sec),  the  wave  reflection  and  transmission  coefficients  of  the  TM  model  agree  well  with  those 
of  the  EB  model  for  both  support  conditions.  However,  as  the  frequency  increases,  the  wave 
propagation  characteristics  of  the  TM  model  differ  significantly  from  those  of  the  EB  model. 
These  differences  can  be  explained  by  examining  the  different  modes  of  vibration.  When 
co>co^  (in  the  regime  of  Case  1),  the  vibrating  motion  of  the  TM  model  is  dominated  by  the  pure 
shearing  motion  [7,  17],  and  hence  the  EB  model,  which  neglects  the  rotary  inertia  and  pure 
shear  effects,  become  inaccurate  at  high  frequency.  As  discussed  in  Ref.  [7],  at  the  finite  cutoff 
frequency,  the  TM  shaft  experiences  no  transverse  displacement,  and  the  cross-section  of  the 
shaft  simply  rotates  back  and  forth  in  unison 

In  Figs.  3(d)-(f)  and  4(d)-(f),  for  =  0  and  £  =  0 ,  the  reflection  and  transmission  coefficients 
of  the  EB  model  are  independent  of  the  frequency.  This  is  because  from  Eqn.  (10a),  A  =  0,  and 
Eqns.  (9c, d)  lead  to  a  single  wavenumber  F,  =  Tj .  From  Appendix  I,  Eqns.  (32*-36*),  the  r  and 
t  are  thus  constant  matrices.  It  is  also  seen  that  the  wave  reflection  and  transmission  coefficients 


1303 


for  both  shaft  models  are  basically  independent  of  the  rotation  speed  over  the  entire  frequency 
range,  even  at  high  rotation  speed  =  0.05  =  44,600  rpm.  In  Ref.  [7],  it  is  also  found  that  has 
negligible  effects  on  the  system  frequency  spectrum,  phase  velocity  and  group  velocity.  On  the 
other  hand,  the  effects  of  the  axial  load  are  significant  for  both  propagating  and  attenuating 
waves  in  the  regime  of  Case  H,  see  Figs.  3(b)-(c)  and  4(b)-(c).  For  both  shaft  models  under 
simple  support  and  compressive  loads  (Figs.  3(b,  e)),  the  reflection  coefficient  ru  of  the  incident 
propagating  wave  is  reduced  significantly  in  the  regime  of  Case  II,  while  the  transmission 
coefficient  fj,  of  the  propagating  wave  component  increases  to  balance  the  energy  carried  in  the 
wave.  However,  the  attenuating  wave  component  which  does  not  carry  any  energy  loses  its 
transmissibility  in  the  same  amount  as  the  reflection  coefficient  r,2 .  Thus,  in  the  presence  of  a 
compressive  load,  most  of  the  transmitted  wave  energy  in  Case  II  comes  from  the  propagating 
component  of  the  incident  wave.  Note  that  axial  tensile  loads  have  the  reverse  effects  on  these 
wave  components.  In  the  clamped  support  case,  the  positive  propagating  wave  component  rn  is 
constant  over  the  regime  of  Case  II  under  any  loading  conditions  for  the  both  shaft  models,  as 
seen  in  Fig.  4. 

Since  there  is  no  damping  at  the  support,  the  incident  power  (Ilinc),  reflected  power  (Hrefi)  and 
transmitted  power  (Iltnin)  in  Cases  II  and  IV  are  related  by  flinc  =  rirefl+ritran  =  Hinc, 

or  Iriil^+Uiil^  =  1.  This  relationship  is  confirmed  by  the  plots  shown  in  Figs.  3  and  4,  where  for 
both  shaft  models,  Irni  and  knl  cannot  exceed  one.  However  in  the  regime  of  Case  I,  in  which 
all  wave  components  propagate,  the  energy  balance  is  Hinc  =  (l^ii+r2iP+lfii+r2!p)  Ilinc,  or  Hinc  = 
(Iri2+r22l^+l?i2+r22l^)  Hinc-  Together  with  the  plots  on  the  phase  of  these  coefficients  (not  shown 
to  minimize  the  size  of  this  manuscript),  the  above  relationships  can  also  be  verified  for  wave 
motion  of  Case  I. 


1304 


support  without  “resistance”.  The  impedance  mismatching  (rn  =  1,  ?,,  =  0)  frequency  at  which 
the  propagating  wave  component  is  completely  reflected  without  being  transmitted  can  also  be 
determined  from  Figs.  4(b,  d)  for  the  two  shaft  models.  This  impedance  mismatching  frequency 
is  located  in  the  regime  of  Case  II  for  the  TM  model  where  the  transverse  mode  dominates  the 
vibrating  motion  of  the  shaft.  Numerical  results  show  that,  as  the  spring  constant  increases,  this 
impedance  mismatching  frequency  increases,  but  is  limited  to  within  the  regime  of  Case  II  and 
can  never  be  found  in  the  regime  of  Case  I  where  the  pure  shearing  mode  dominates  the  vibrating 
motion  of  the  shaft  (refer  to  Fig.  3  for  the  transition  of  types  of  wave  motion). 

Figure  6  shows  the  reflection  and  transmission  coefficients  for  waves  incident  upon  a  support 
having  both  translational  and  rotational  constraints.  Since  both  flexural  and  shearing  modes  of 
vibration  are  constrained  at  this  support,  the  maximum  of  the  reflection  coefficient  is  expected  to 
be  higher  than  the  previous  case.  Figures  6(a-b)  and  (c-d)  are  the  results  for  the  TM  and  EB 
models,  respectively.  The  translational  and  rotational  spring  constants  used  in  the  simulations 
are  k,Q-  10^  N/m  and  ^^5=  10^  Nm/rad,  respectively.  It  is  noted  that  in  the  regime  of  Case  II,  i.e.. 


Figure  6.  Wave  reflection  and  transmission  coefficients  at  an  elastic  support  with  translational  and  rotational  springs 
{k,-k,Q,  k^^O,  c,  =  c,  =  m=  y,„  =  0)  as  a  function  of  frequency,  J3  =  0.05  and  e  =  0.  (a-b)  and  (c-d)  are  results 
for  the  Timoshenko  and  Euler-Bernoulli  shaft  models,  respectively. 


modulus  modulus 


2.0 


r,2  (upper  line) 

r,2=:r2,  (middle  dashed  line) 

Tj,  (lower  line) 


Figure  4.  Wave  reflection  coefficients  at  a  clamped  support  (/c,  =  fc,  =  ■»  and  =  c,  =  m  =  =  0)  as  a  function  of 

frequency,  (a)-(c)  and  (d)-(f)  are  the  results  for  the  Timoshenko  and  Euler-BernouUi  shaft  models,  respectively.  The 
transition  from  one  type  of  ivave  motion  to  another  is  marked  for  the  case  /J  =  0.05 ,  £  =  -0.05 . 


3.2,  Wave  reflection  and  transmission  at  elastic  supports 

Figure  5  shows  the  reflection  and  transmission  coefficients  for  waves  incident  upon  a  support 
with  a  finite  translational  spring  for  three  different  spring  constants.  Figures  5(a)-(b)  and  (c)-(d) 
are  results  for  the  TM  and  EB  models,  respectively.  The  spring  constant  used,  k^Q  =  10^  N/m,  is 
a  typical  bearing  spring  constant  value  for  turbine  generators.  The  plots  show  that  there  is  no 
significant  difference  in  the  moduli  between  the  two  shaft  models.  This  is  because  the  incident 
wave  does  not  experience  any  rotational  constraint  at  the  support,  and  hence  the  additional  rotary 
inertia  factor  in  the  TM  model  has  only  a  small  contribution  to  the  wave  motions.  As  the  support 
spring  constant  increases,  the  curves  for  both  the  reflection  and  transmission  coefficients  are 
shifted  to  the  right  and,  as  the  spring  constant  approaches  infinity,  these  curves  eventually 
become  asymptotic  to  those  shown  in  Fig.  3.  Note  that  an  impedance  matching  (r  =  0,  t  =  I), 
where  all  wave  components  are  transmitted  without  being  reflected,  is  found  in  the  high 
frequency  region  for  both  shaft  models.  Thus,  as  the  frequency  increases,  the  characteristics  of 
waves  travelling  along  the  shaft  remain  unchanged  such  that  waves  propagate  through  this  elastic 


Figure  5.  Wave  reflection  and  transmission  coefficients  at  an  elastic  support  with  a  translational  spring 

=  c,  =  c,  =m  =  =0)  as  a  function  of  frequency,  /3  =  0.05  and  £  =  0.  (a-b)  and  (c-d)  are  results  for  the 

Timoshenko  and  Euler-Bernoulli  shaft  models,  respectively. 


1308 


in  the  low  frequency  range,  both  shaft  models  have  similar  reflection  characteristics,  and  both  the 
reflection  and  transmission  coefficients  are  not  significantly  affected  by  the  rotational  spring. 
However,  as  the  frequency  increases,  the  effect  of  the  rotational  constraint  on  the  wave  motion 
becomes  eminent,  particularly  for  the  TM  model.  As  seen  in  Figs.  6(a)  and  6(c),  the  reflection  of 
the  attenuating  wave  components  are  significantly  higher  than  those  of  the  propagating  wave 
components.  Hence,  when  a  rotating  shaft  has  a  clamped  support(s)  such  as  a  journal  bearing, 
contributions  from  the  attenuating  wave  components  should  be  included  in  the  formulation  since 
a  significant  amount  of  energy  in  the  propagating  component  arises  from  the  incident  attenuating 
wave  component.  It  is  noted  that  the  impedance  matching  regions  seen  in  Figs  5(a,  c)  disappear 
when  the  rotational  constraint  is  added.  Moreover,  the  impedance  mismatching  frequency  shown 
in  Figs.  5(b,  d),  which  is  found  in  the  regime  of  Case  II,  also  does  not  occur.  At  low  frequency  in 
Figs.  6  (b,  d),  there  appears  to  be  a  mismatching  region,  but  rj,  is  not  exactly  equal  to  zero. 
From  Figs.  6(a,  c),  it  is  seen  that  there  is  a  frequency  at  which  the  positive  propagating  wave 
component  r,,  is  zero  (this  frequency  is  slightly  different  for  the  two  models).  This  frequency 
does  not  correspond  to  an  impedance  matching,  though  the  propagating  wave  is  not  reflected  at 
all  but  is  only  transmitted  (r,i  =  1).  Based  on  other  research  results  [18],  this  phenomenon  likely 
indicates  a  structural  mode  delocalization  in  bi-coupled  systems,  in  which  vibrations  on  both 
sides  of  the  support  become  strongly  coupled.  Further  research  on  the  vibrations  of  rotating 
shafts  with  intermediate  supports  is  being  pursued  to  confirm  the  mode  delocalization. 

Figure  7  plots  the  effects  of  axial  compressive  loads  on  the  wave  reflection  and  transmission 
upon  a  support  with  finite  spring  constant  for  the  Timoshenko  shaft  model.  As  seen  in  Fig.  7(a), 
the  reflection  coefficient  for  the  incident  propagating  wave  component  ru  is  substantially 
reduced  in  the  low  frequency  range  while  the  reflection  coefficient  for  the  incident  attenuating 
wave  component  increases  significantly.  However,  Fig.  7(b)  shows  the  reversed  effects  on  the 
transmission  coefficient.  It  can  therefore  be  concluded  that,  when  the  shaft  is  axially  strained  by 


Figure  7.  Wave  reflection  and  transmission  coefficients  at  an  elastic  support  (A:^  =  k,Q  and  kr  =  c,  =  Cr  =  m  =  J„,  =  0) 
for  the  Timoshenko  shaft  model  with  and  without  the  compressive  load,  (a)  reflection  coefficients,  (b)  transmission 
coefficients. 


1309 


Figure  8.  Wave  reflection  and  transmission  coefficients  at  an  elastic  support  k^  =  and  c,=  Cr  =  m  = 

J„,  =  0)  for  the  Timoshenko  shaft  model  with  and  without  the  compressive  load,  (a)  reflection  coefficients,  (b) 
transmission  coefficients. 


compressive  loads,  the  energy  contribution  from  the  incident  attenuating  wave  component  to  the 
energy  in  the  reflected  propagating  wave  is  more  significant  than  the  strain-free  situation  in  the 
low  frequency  range,  while  most  of  the  energy  in  the  transmitted  wave  derives  from  the  incident 
propagating  wave  component. 

Figure  8  plots  the  wave  reflection  and  transmission  coefficients  along  an  axially  compressed 
Timoshenko  shaft  model  at  a  support  with  finite  translational  and  rotational  spring  constants. 
Similar  results  to  the  previous  example  can  be  observed  in  terms  of  energy  contribution  from  the 
incident  attenuating  wave  component  in  the  low  frequency  range.  However,  the  effects  of  the 
axial  compressive  load  on  both  the  reflection  and  transmission  coefficients  for  the  propagating 
wave  component  ( r^^  and  r,, )  are  significantly  reduced  when  compared  to  Fig.  7. 


3.3.  Wave  reflection  and  transmission  at  damped  supports 

Figure  9  shows  the  effects  of  both  translational  and  rotational  dampers  at  a  support  with  finite 
translational  and  rotational  spring  constants.  Figures  9(a,  b)  and  (c,  d)  are  results  for  the  TM  and 
EB  models,  respectively.  The  translational  and  rotational  damping  constants  used  in  this  study 
are  c^to  =  2x10^  Ns/m  and  Cdro  =  64x10^  N-m-s/rad,  typical  values  for  bearings  in  turbine 
generators.  The  curves  with  symbols  (•  and  ♦)  are  the  results  when  the  rotational  damping  factor 
is  also  included  in  the  formulation.  It  can  be  seen  that  I  and  Ir^l  for  both  shaft  models  are 
significantly  lowered  due  to  the  presence  of  damping.  Note  that,  because  of  the  damping,  the 
frequency  at  which  1  rj,  I  =  0  (compare  with  Figs.  6(a,  c))  no  longer  exists  for  both  shaft  models.  It 
can  also  be  seen  that  the  effect  of  the  rotational  damping  factor  on  the  wave  reflection  and 
transmission  is  not  significant  over  the  entire  frequency  range  for  both  shaft  models.  For  TM 
model,  the  contribution  of  the  rotational  damping  to  both  Ir;,  1  and  1?,,  I  is  almost  negligible.  The 


1310 


support  condition  considered  in  this  particular  example  is  simulated  as  an  actual  bearing  support 
adopted  in  turbine  generators.  Hence  for  this  particular  type  of  bearing  support,  the  effect  of  the 
rotational  damping  on  wave  reflection  and  transmission  is  not  considerable.  Other  numerical 
results  (not  shown  in  this  paper)  show  that  the  wave  propagation  at  the  damped  support  is 
characterized  by  translational  damping  rather  than  rotational  damping.  Note  that  similar  results 
have  been  presented  for  the  support  without  damping  (see  Fig.  6). 


Figure  9.  Wave  reflection  and  transmission  coefficients  at  an  elastic  support  with  damping  (k,  =  k,o,  K  =  Ko,  c,  =  c,;,o, 
c,  =  cjro  and  m  =  J,„  =  0)  as  a  function  of  frequency  for  /3  =  0.05  and  £  =  0.  (a-b)  and  (c-d)  are  results  for  the 
Timoshenko  and  Euler-Bernoulli  shaft  models,  respectively. 


3.4.  Wave  reflection  and  transmission  at  a  rotor  mass 

Consider  a  gear  rigidly  assembled  to  a  rotating  shaft.  The  gear  is  assumed  to  be  perfectly 
balanced  and  its  thickness  is  sufficiently  small  such  that  wave  reflection  and  transmission  due  to 
the  geometric  discontinuity  between  the  shaft  and  the  gear  can  be  neglected.  However  the  gear 
does  resist  the  translational  and  rotational  motions  of  the  cross-sectional  element  of  the  shaft. 


1311 


Figure  10  shows  the  reflection  and  transmission  upon  the  gear  when  the  mass  mo  and  mass 
moment  of  inertia  J^o  of  the  gear  are  4  and  16  times  of  the  shaft,  respectively.  Not  shown  in 
Figs.  10  (b,  d)  is  that  r,2  =  0  when  w=  /„  =  0.  Like  some  previous  support  conditions 

discussed,  the  effects  of  the  rotor  mass  are  much  more  significant  in  the  high  frequency  region 
for  both  models  (particularly  around  and  beyond  the  cutoff  frequency  for  the  TM  model).  In 
general,  the  rotor  mass  decreases  the  transmission  and  increases  the  reflection  of  the  wave.  At 
very  high  frequency,  there  is  basically  no  wave  transmission.  Note  that,  since  the  geometric 
discontinuity  between  the  shaft  and  the  gear  is  neglected  in  this  model,  one  may  expect  that  the 
actual  reflection  for  both  the  propagating  and  attenuating  wave  components  would  be  higher. 


(a) 


(m=  =  V 


(b) 


- (m  =  0, 

- (  m  =  mg .  ) 


/ 


Figure  10.  Wave  reflection  and  transmission  at  a  rotor  mass  assembled  to  a  rotating  shaft  (*,  =  0  =  L  =  Cf  =  Cr  =  0, 
and  m  =  rriQ,  and  J,„  =  J„^)  as  a  function  of  frequency  when  P  =  0.05  and  £  =  0.  (a-b)  and  (c-d)  are  results  for  the 
Timoshenko  and  Euler-Bernoulli  shaft  models,  respectively. 


4.  WAVE  REFLECTION  AND  TRANSMISSION  AT  A  GEOMETRIC  DISCONTINUITY 
It  is  common  for  a  rotating  shaft  element  to  have  changes  in  cross-section,  or  to  be  joined  to 


1312 


Case  I,  II,  or  IV 

A,  =  (1  +  a)co-  -  2j3co  -  16£  (1  +  £  -  ~) 

a 


Z=0 


Case  I,  II,  or  IV 


B,  =  co' 


ccco-  -  2p(o  -  I6a  (1  +  £)(1  +  £ - ) 

a 


A={\  +  a)(o^-2^co-^^^  (l  +  £,  -— ) 
o"  a 

16a 


=  co^\ a  (O'  -215(0-^^  (l+£j(l  +  £  -— ) 

L  O’  a 


Figure  11.  Wave  reflection  and  transmission  at  a  geometric  discontinuity. 


another  shaft  element  by  a  coupling.  Figure  1 1  shows  a  typical  example  of  a  discontinuous  shaft 
model  in  which  two  shafts  of  differing  wavenumber  and  diameter  are  joined  at  z  =  0 .  The 
subscripts  I  and  r  denote  z  =  O'  and  z  =  0^  regions,  respectively.  It  is  known  that  when  a  wave 
encounters  a  junction  or  a  discontinuity,  its  wavenumber  is  changed.  It  is  therefore  possible  that 
a  wave  on  the  left  side  of  the  junction  can  be  propagating,  while  after  crossing  the  junction  to  the 
right  side,  the  wave  becomes  attenuating.  Therefore,  for  a  Timoshenko  shaft,  when  a  wave 
propagates  through  the  junction,  there  are  mathematically  nine  possible  different  combinations  of 
wave  motions  to  be  considered  depending  on  the  values  of  the  functions  A  and  B  on  each  side  of 
the  junction,  as  depicted  in  Fig.  12. 


Figure  12.  Nine  possible  combinations  of  wave  motions  at  a  geometric  discontinuity  of  the  cross  section  for  the 
Timoshenko  shaft  model.  Subscripts  /  and  r  denote  the  left  and  the  right  side  of  the  discontinuity,  respectively. 


1313 


For  simplicity,  assume  that  material  properties  such  as  p,  E,  and  G  are  the  same  for  both  sides 
of  shaft  element.  The  displacement  continuity,  moment  and  force  equilibrium  conditions  are 
applied  at  the  junction  to  determine  the  wave  refection  and  transmission  matrices.  Results  for  the 
three  most  commonly  encountered  possibilities  in  the  low  frequency  regime  are  listed  as  follows. 

Case  II  {A,  >0,  Bi  <0)  -  Case  I  (A^  >  0 ,  R,  >0): 


■  1  r 

1 

1  ' 

r  1  M 

rC"  = 

tCY 

Jlu  ^21 . 

nu  ■ 

~n2i. 

Jl\r  ^2r_ 

r  -i^Bu 

^21^2! 

C"  + 

r  Xtnu 

”Y,r72,  1 

L^xr., 

-riu)  r 

21 -in  21^ 

1 

Y 

rr 

1 

T 

^772,)J 

rC-^ 


Yuriu  72rn2r 

L-io-"(r„-77„)  -i<y^(r2r-V2r)J 


tc", 


Case  II  (A,  >0,  B,  <0)  -  Case  II  (A^  >  0 ,  B^<0): 


■  1  1  ■ 

■  1  1  ■ 

■  1  1  * 

C"  + 

rC^  = 

Jlv  n2l. 

—  nu  “^2/. 

Y.r  n2r. 

(37a) 


(37b) 


(38a) 


-"^21^21 

^2,  -in 21 


C"  + 


i^unu  ^21^21 

~i(^H  —  nu  )  ~(^2/  ~  ^^21  ) 


rC" 


-i<yXrnu  -^Xr'n2r 
-zcr"(r,,-77„)  -C7“(r2,-7]2,)J 


CaseII{A^>0,  B,  <0)  -  Case  IV  {A,  <0 ,  S,  <0): 


r  i  1  1 

r  1  1 1 

■  1  r 

C"  + 

rC^  = 

nu  n2i 

l-nu  -n2i^ 

n2r  n^r. 

Y/^2/ 

^2/^2/ 

JXi—nu)  Y/~^*^2;. 

“(Yf  -in2i)_ 

-i^%rn2r 


-<yXrn:r 


tc\ 


(38b) 


(39a) 


(39b) 


where  cr  is  the  diameter  ratio  between  the  shaft  elements,  defined  as 


1314 


(40) 


CJ  =  — . 


Note  that  77/5  in  Eqns.  (37a-39b)  are  given  by  Eqns.  (20a,  b),  (22a,  b),  and  (24a,  b)  according  to 
the  type  of  wave  motion,  and  77/  s  on  the  right  side  of  the  geometric  discontinuity  are  modified 
as  follows. 


where 


:  -  77  = - —  for  Case  I, 

r„e;  yzX 

(41a,  b) 

-co^  Tl+co- 

,  Ti,  -  — - 7-  for  Case  II, 

r„e;  iT^rS; 

(42a,  b) 

,7]2  for  Case  IV, 

ir,x  r,,e; 

(43a,  b) 

£'  =  1  +  8,.  — and  £,  =  —  . 

(44) 

a  O' 


Moreover,  the  wavenumbers,  A  and  B  of  the  shaft  element  on  the  right  side  of  the  junction  are 
modified  as  follows. 

y „  =  ^[a,  +  ^Aj-4B^f .  r,,  =  -  Va?-4B,)^  ,  (45a.  b) 

+  =  .  (45c, d) 

where, 

A.  =  (l  +  a)a.=  -2pa)-^a  +  £,-|), 


B.  =co‘ 


am--2Pa-^  (l  +  £,)(l  +  £,-^) 

C  OL 


(46b) 


Corresponding  results  for  the  simple  Euler-Bemoulli  shaft  model  are  listed  in  Appendix  II. 

Figures  13  to  16  show  some  representative  examples  of  wave  reflection  and  transmission 
upon  the  geometric  discontinuity.  In  Figs.  13  and  14,  the  thick  and  thin  curves  represent  results 


1315 


for  the  TM  and  EB  models,  respectively.  The  second  graph  in  each  figure  shows  the  changes  of 
Ai,  Bi,  Ar  and  Br,  and  how  wave  solutions  on  both  sides  of  the  discontinuity  change  as  the 
frequency  increases  for  the  TM  model.  In  general,  the  wave  reflection  and  transmission  for  the 
EB  model  are  frequency  independent  except  when  the  shaft  is  axially  strained,  while  the  wave 
propagation  characteristics  for  the  TM  model  are  strongly  dependent  on  the  frequency. 

Comparing  Figs.  13  and  14,  it  is  noted  that,  for  both  shaft  models,  the  average  reflection  and 
transmission  rates  for  cr  =  0.8  are  higher  than  those  for  o'  =  1.2 ,  especially  for  the  attenuating 
wave  components.  These  results  imply  that  incident  attenuating  waves  contribute  more  energy  to 
propagating  waves  at  the  discontinuity  when  the  waves  travel  from  a  smaller  to  a  larger  cross- 
section.  In  particular,  it  is  noted  that  the  transmissibiiity  of  the  attenuating  wave  tn  has  a  strong 
dependency  on  the  direction  of  propagation.  Note  also  that  the  differences  between  the  two  shaft 
models  are  more  pronounced  when  cr  =  0.8 .  It  is  clearly  seen  from  the  figures  that  when  Bj  and 
B^  change  from  negative  to  positive,  both  reflection  and  transmission  coefficients  experience  a 
sharp  jump  or  drop  at  the  finite  cutoff  frequencies,  due  to  changes  in  the  types  of  wave  motion. 
In  the  frequency  region  (S,  >  0  and  B^  <0)  located  between  the  two  cutoff  frequencies  in  Fig. 


2.0  h 


0.5 


Thick  curves  :  Timoshenko  shaft  model 
Thin  curves  :  Euier-Bernoulli  shaft  model 


- Ui 


h 

i 

il _ 


- 

7/  X. - V' 


.'7 


\ 


5 

(O 


Figure  13.  Reflection  and  transmission  of  waves  incident  upon  a  change  in  the  cross-section,  a=  0.8,  P  =  0.05,  and 
£  =  0.  Thick  and  thin  curves  are  results  for  the  Timoshenko  and  Euier-Bernoulli  shaft  models,  respectively.  Note 
that  the  ordinates  in  the  lower  graphs  keep  increasing  with  frequency  (abscissa). 


1316 


13,  the  wave  motion  on  the  left  side  of  the  junction  is  governed  by  the  wave  solution  of  Case  I 
since  all  wave  components  are  propagating  at  a  frequency  larger  than  the  cutoff  frequency,  while 
the  wave  motion  on  the  right  side  of  the  junction  is  governed  by  the  wave  solution  of  Case  II. 
Thus,  for  O’  =  0.8 ,  some  of  the  propagating  wave  components  on  the  left  side  of  the  shaft  element 
cannot  propagate  as  they  pass  the  discontinuity,  and  become  attenuating.  A  similar,  but  converse 
conclusion  can  be  drawn  for  the  frequency  region  (5^  >  0,  5,  <  0)  when  cr  =  1.2 ,  as  shown  in 
Fig.  14.  The  results  of  Figs.  13  and  14  show  that,  for  different  system  parameters  cr,  jS,  and  e  and 
at  any  given  frequency,  the  types  of  wave  motion  on  each  side  of  the  discontinuity  can  be 
different,  as  depicted  in  Fig.  12. 

From  Eqns.  (41a-43b),  it  is  seen  that  when  the  Timoshenko  shaft  is  axially  strained  and  (O  is 
not  sufficiently  large,  the  wavenumber  (hence  wave  propagation  characteristics)  depends  strongly 
on  the  cross-section  ratio  <7.  Figure  15  shows  the  effects  of  the  axial  load  on  the  wave  reflection 
and  transmission,  which  are  mostly  limited  to  the  relatively  low  frequency  region.  In  Figs.  15(a- 
b),  when  the  shaft  is  axially  compressed  (£  =  -0.05),  the  reflection  and  transmission  due  to  the 
incident  attenuating  wave  component  decrease  for  both  o  <  1  (plot  (a))  and  <7  >  1  (plot  (b)). 
However,  the  transmission  due  to  an  incident  propagating  wave  decreases  significantly  for 


—  —  f,  2 


2.0  h 


Figure  14.  Reflection  and  transmission  of  waves  incident  upon  a  change  in  the  cross-section,  cr=  1.2,  /3  =  0.05,  and 
£  =  0.  Thick  and  thin  curves  are  results  for  the  Timoshenko  and  Euler-Bernoulli  shaft  models,  respectively.  Note 
that  the  ordinates  in  the  lower  graphs  keep  increasing  with  frequency  (abscissa). 


1317 


Figure  15.  Reflection  and  transmission  of  waves  upon  a  change  in  the  cross-section  when  =  0.05  for  the 
Timoshenko  shaft  model,  (a)  <7=0.8  and  £  =  -0.05.  (b)  (T=  1.2  and  e= -0.05.  (c)  cr=  0.8  and  £=  0.05.  (d)  or  =  1.2 
and  e=  0.05.  Thin  and  thick  curves  show  the  results  when  the  shaft  is  strain-free  (£=  0)  and  strained,  respectively. 


c  -  0.8  and  increases  for  <7  =  1.2  at  low  frequency. 

Effects  of  the  axial  load  on  the  wave  reflection  and  transmission  are  more  significant  when 
the  shaft  is  compressed  (Figs.  15(a-b))  than  when  it  is  under  tension  (Figs.  15(c-d)).  This  is 
because  the  wavenumbers  of  both  the  propagating  and  attenuating  wave  components  are  only 
slightly  changed.  It  is  also  noted  that,  in  the  low  frequency  range,  the  wave  solution  of  Case  IV 
governs  the  wave  motions  on  both  sides  of  the  discontinuity,  and  the  wave  components  which 
have  large  wavenumber  (Fi)  attenuate,  while  wave  components  with  small  wavenumber  (r2) 
propagate  along  the  waveguide  as  long  as  A  remains  negative. 


1318 


5.  WAVE  REFLECTION  AT  BOUNDARIES 


When  a  wave  is  incident  upon  a  boundary,  it  is  only  reflected  because  no  waveguide  exists 
beyond  the  boundary.  Consider  an  arbitrary  boundary  condition  with  translational  and  rotational 
spring  constraints,  dampers,  and  a  rotor  mass,  as  shown  in  Fig.  16.  The  reflection  matrix  at  the 
boundary  is  derived  for  each  Case.  Applying  the  same  non-dimensional  parameters  employed  in 
Section  3,  and  by  imposing  the  force  and  moment  balances  at  the  boundary,  which  can  be 
deduced  by  eliminating  and  inEqns.  (27a,  b), 

M~  =  k^y/  +  c^yr  +  J„y/ ,  (47  a) 

-V  =  k,u  +  c,u  +  mu ,  (47b) 

the  reflection  matrix  for  each  Case  is  determined. 

CaseI{A>0,  B>0): 

J  r72(ir2-2:,J  T’r-T7,(jri  +  ^.)  + 

+  Kr2-Bi)  +  ^s\  Kr2-T72)-^.J’ 


Case  7/  ( A  >  0 ,  5  <  0): 

7],(ir,-2:j  T'r-77,(ir,+Ej  -7]2(r2  +  zj1 

'''[i(r,-r7,)+z,  (r,-ii7,)+i.J  [i(r,-j?,)-z, 


M,Jm 


Z=0 


Figure  16.  Wave  reflection  upon  a  general  boundary. 


1319 


CaseIV{A<0,  5<0): 


772  (^r2  s„,) 

iCFj  -  7)2)  + 


(r.-mj+s. 


772  (zTj  +  Sp,) 

/(r2  ~  ^2)  ■“ 


-^iCr.+zj 

(r,-j77,)-E, 


(50) 


where  77's  in  above  equations  have  been  defined  in  Eqns.  (20a,  b),  (22a,  b)  and  (24a,  b),  and 

Z„,  =  +  ic^co  -  J,„co~ ,  and  =  ^,  +  ic,Q)  -  mco^  ,  (51a,  b) 

The  corresponding  results  for  the  simple  Euler-Bernoulli  shaft  model  are  listed  in  the  Appendix 
in.  By  specifying  the  parameters  in  the  reflection  matrix  r,  results  for  three  typical  boundary 
conditions  (simple  support,  clamped  support,  and  free  end)  can  be  obtained. 

•  Simple  support  {k,  =^,  =m  =  c,  =  =  7,„  =  0) 


r 


-1  0 
0  -1 


for  Case  I,  II,  and  IV, 


(52) 


•  Clamped  support  (k^  =  k,.  =  m  =  c,  =  =  J„,  =  0) 

1 

Tli-ri2 


(53b) 

(53c) 

•  Free  end  {k,  =  k^  =  m  =  c,  =  =  /„,  =  0 ,  and  £  =  0) 


1 


rii+ri2 


2772 


77,-  772  L-277i  -(r?, +772)J 

1 


-irii+Tli)  "2772 

2t},  (771+772) 


for  Case  II, 


for  Case  IV, 


■77, +7?2  2772 

-277,  -(771+772), 


for  Case  I, 


(53a) 


r 


J_ 


77,7?2(ri+72)”7ir2(77, +772) 
-277,7i(77, -7i) 


277272(772-72) 

-77,772(71 +72)  +  7i72(77i +772). 


(54a) 


where.  A,  =  r],n,  -  7, )  +  7,72 (77,  -  ^2 )  for  Case  I, 


+72)  +  7i72(77i  +772)  2i7}^y^iin^ -y^) 

^  A;,  [  2/77,7,(771 -7,)  77,772(/7, +72)-7i72(7?, +772) 


(54b) 


where,  =77,772 (77,  -72) -7,72(7?,  - 772)  for  Case//, 


1320 


Figure  17.  An  example  of  a  rotating  shaft  with  multiple  supports  and  discontinuities. 


I  \TiiV2(ri+ir2)-rj2(Tii+r]2)  277,7,(77, +ir,)  '  ^ 

r  = -  (54c) 

A/v  L  -2i7?2r,(77j-7j  -77,772(7,  +J72)+rir2(^i  +^2). 

where,  A;^  =77,772(71  -172)  “7172 (Hi -772)  for  Case  IV. 

6.  APPLICATIONS 

The  reflection  and  transmission  matrices  for  waves  incident  upon  a  general  point  support  or  a 
change  in  cross-section  can  be  combined  with  the  transfer  matrix  method  to  analyze  the  free 
vibration  of  a  rotating  Timoshenko  shaft  with  multiple  supports  and  discontinuities,  and  general 
boundary  conditions.  The  basic  idea  of  this  technique  has  been  shown  in  Ref.  [6].  However,  due 
to  the  complex  wave  motions  in  the  Timoshenko  shaft  model,  such  as  the  frequency  dependency 
of  the  wave  reflection  and  transmission  at  a  cross-section  change,  it  is  important  to  apply  the 
proper  reflection  and  transmission  matrices  consistent  with  the  values  of  A  and  B  on  both  sides  of 
the  discontinuity,  particularly  when  numerical  calculations  are  performed.  Consider  for  example 
the  free  vibration  problem  of  the  rotating  Timoshenko  shaft  model  shown  in  Fig.  17.  Denoting  R 
as  a  reflection  matrix  which  relates  the  amplitudes  of  negative  and  positive  travelling  waves  at  a 
discontinuity,  and  defining  T/  as  the  field  transfer  matrix  which  relates  the  wave  amplitudes  by 

C^iZo  +  z)  =  TC"(Zo) ,  C-(zo  +  z)  =  T-'C-(zo) ,  (55) 

the  following  relations  can  be  found. 


1321 


W-=R5W^ 

(R5=>-s). 

(56a) 

i  =  2,3,4  (station  number), 

/i  =  left  (/)  or  right  (r) 

(56b) 

w>T,w-,, 

(56c) 

<  =r,w-, 

(56d) 

<=T.w;, 

(56e) 

where  in  Eqn.  (56b), 

R,,=TiR„,,T,.  R,,=r,+t,(R-’-r,-)-'t,  (56f) 


Solving  the  above  matrix  equations  gives 

(r,T,R„T,-I)w>0,  (57) 

where  each  element  of  the  matrix  is  a  function  of  two  different  wavenumbers  and  the  frequency 
CO.  For  non-trivial  solutions,  the  natural  frequencies  are  obtained  from  the  characteristic  equation 

Det[(r,T,R„T,-I)]  =  0.  (58) 

The  proposed  method  is  applied  to  an  example  of  a  two-span  rotating  shaft,  simply  supported 
at  the  ends  and  with  an  intermediate  support  consisting  of  translational  and  rotational  springs,  as 
shown  in  Fig.  18.  Numerical  computations  were  performed  by  a  PC-based  Mathematical.  The 
values  of  the  spring  constants  Kj  and  Kr  are  those  introduced  in  Section  3.2,  with  ^  =  1  m  and 
the  rotation  parameter  p  =  0.05 . 


1322 


Figure  19  shows  the  first  eight  natural  frequencies  of  the  vibrating  shaft  for  both  the  TM  and 
EB  models  for  a  classical  simple  intermediate  fixed  support  K^=0)  placed  at 

various  locations.  The  results  confirm  the  well-known  fact  that  the  Timoshenko  model  leads  to 
smaller  eigenvalues.  Figure  20  shows  the  first  eight  natural  frequencies  of  the  Timoshenko  shaft 
for  an  elastic  intermediate  support  with  three  different  translational  spring  constants.  It  can  be 
seen  that  the  effect  of  the  translational  spring  diminishes  for  higher  modes.  The  proposed  wave 
analysis  technique  can  also  be  applied  effectively  to  the  study  of  structural  mode  localizations  in 
mistuned,  rotating  systems.  Dynamics  of  such  systems  will  be  addressed  in  another  paper. 


Figure  19.  Natural  frequencies  of  a  two-span,  rotating 
shaft  as  a  function  of  the  support  location;  intermediate 
support  is  fixed. 


Figure  20.  Natural  frequencies  of  a  two-span  rotating 
Timoshenko  shaft  as  a  function  of  the  support  location; 
intermediate  support  consists  of  k,  and  kr . 


7.  SUMMARY  AND  CONCLUSIONS 

In  modern  high  speed  rotating  shaft  applications,  it  is  common  that  the  shaft  has  multiple 
intermediate  supports  and  discontinuities  such  as  bearings,  rotor  masses,  and  changes  in  cross- 


1323 


sections.  In  many  cases,  the  ratio  of  the  shaft  diameter  to  its  length  between  consecutive  supports 
is  large,  and  the  Timoshenko  model  (TM)  is  needed  to  accurately  account  for  the  shear  and  rotary 
inertia  effects.  In  this  paper,  the  wave  propagation  in  a  rotating,  axially  strained  Timoshenko 
shaft  model  with  multiple  discontinuities  is  examined.  The  effect  of  the  static  axial  deformation 
due  to  an  axial  load  is  also  included  in  the  model.  Based  on  results  from  Ref.  [7],  there  are  four 
possible  types  of  wave  motions  {Cases  I,  II,  III  and  IV)  in  the  Timoshenko  shaft,  as  shown  by 
Eqns.  (5a-8b).  In  practice.  Case  III  does  not  occur  and  is  excluded  in  the  analysis.  For  each 
Case,  the  wave  reflection  and  transmission  matrices  are  derived  for  a  shaft  under  various  support 
and  boundary  conditions.  Results  are  compared  with  those  obtained  by  using  the  simple  Euler- 
Bemoulli  model  (EB)  and  are  summarized  as  follows. 

1)  In  general,  the  two  shaft  models  show  good  agreement  in  the  low  frequency  range  where  the 
wave  motion  is  governed  by  Case  II  and  Case  IV.  However,  at  high  frequencies,  the  types  of 
wave  motions  and  propagation  characteristics  for  the  TM  and  EB  models  are  very  different. 

2)  The  effects  of  shaft  rotation  on  the  wave  reflection  and  transmission  are  negligible  over  the 
entire  frequency  range  and  even  at  high  speed  (up  to  44,600  rpm).  While  the  effects  of  the 
axial  load  are  significant,  especially  in  the  low  frequency  range. 

3)  When  waves  are  incident  at  supports  with  only  translational  springs,  differences  in  the  results 
between  the  TM  and  EB  models  are  small,  and  there  exists  frequency  regions  of  impedance 
matching  and  an  impedance  mismatching  frequency  (limited  to  within  the  regime  of  Case  IT). 
The  impedance  matching  and  mismatching  disappear  when  a  rotational  spring  is  added  to  the 
support.  Instead,  there  is  a  frequency  at  which  Irul  =  0  and  Ifni  =  1,  and  vibrations  on  both 
sides  of  the  support  become  strongly  coupled.  This  (delocalization)  phenomenon  suggests 
further  research  on  the  vibrations  of  constrained  multi-span  beams.  When  there  is  damping 
at  the  support,  the  frequency  at  which  IrnI  =  0  does  not  occur.  Moreover,  effects  of 
translational  damping  on  the  wave  propagation  are  more  significant  at  high  frequency, 
especially  for  the  TM  model,  however  effects  of  rotational  damping  is  not  significant  over  the 
entire  frequency  range. 

4)  Contributions  of  attenuating  wave  components  to  the  energy  in  the  reflected  and  transmitted 
waves  are  significant  when  the  shaft  is  axially  strained  and  when  the  support  has  a  rotational 
constraint.  Thus  attenuating  waves  should  be  included  in  the  formulation. 

5)  Unlike  the  spring  supports,  in  which  waves  are  easily  transmitted  at  high  frequency,  the  rotor 
mass  support  diminishes  the  wave  transmission  as  the  frequency  increases. 

6)  When  waves  are  incident  at  a  geometric  discontinuity  such  as  a  change  in  the  cross-section, 
there  are  nine  possible  combinations  of  wave  motions  on  both  sides  of  the  discontinuity.  It  is 
shown  that  differences  of  the  results  between  the  TM  and  EB  models  depend  on  the  diameter 
ratio  (and  hence  the  direction  of  the  wave  incidence).  Moreover,  incident  attenuating  waves 
contribute  more  energy  to  propagating  waves  at  the  discontinuity  when  the  waves  travel  from 
a  smaller  to  a  larger  cross-section.  When  the  shaft  is  axially  strained,  the  effects  of  the  load 
on  the  wave  propagation  are  primarily  limited  to  the  low  frequency  range. 

The  reflection  and  transmission  matrices  are  combined  with  the  transfer  matrix  method  to 
provide  a  systematic  solution  method  to  analyze  the  free  vibration  of  a  multi-span,  rotating  shaft. 
Since  the  procedure  involves  only  2x2  matrices  (while  including  the  near-field  effects  already), 
strenuous  computations  associated  with  large-order  matrices  are  eliminated. 


1324 


ACKNOWLEDGMENTS 


The  authors  wish  to  acknowledge  the  support  of  the  National  Science  Foundation  and  the 
Institute  of  Manufacturing  Research  of  Wayne  State  University  for  this  research  work. 


REFERENCES 

1.  Lin,  Y.K.,  Free  Vibrations  of  a  Continuous  Beam  on  Elastic  Supports.  International  Journal 
of  Mechanical  Sciences,  1962,  4,  pp.  409-423. 

2.  Graff,  K.F.,  Wave  Motion  in  Elastic  Solids,  Ohio  State  University  Press,  1975. 

3.  Cremer,  L,,  Heckl,  M.  and  Ungar  E.E.,  Structure-Bome  Sound,  Springer-Verlag,  Berlin, 
1973. 

4.  Fahy,  F.,  Sound  and  Structural  Vibration,  Academic  Press,  1985. 

5.  Mead,  D.J.,  Waves  and  Modes  in  Finite  Beams:  Application  of  the  Phase-Closure  Principle. 
Journal  of  Sound  and  Vibration,  1994, 171,  pp.  695-702. 

6.  Mace,  B.R.,  Wave  Reflection  and  Transmission  in  Beams.  Journal  of  Sound  and  Vibration, 
1984,  97,  pp.  237-246. 

7.  Kang,  B.  and  Tan,  C.A.,  Elastic  Wave  Motions  in  an  Axially  Strained,  Infinitely  Long 
Rotating  Timoshenko  Shaft.  Journal  of  Sound  and  Vibration  (submitted),  1997. 

8.  Han,  R.P.S.  and  Zu,  J.W.-Z.,  Modal  Analysis  of  Rotating  Shafts:  A  Body-Fixed  Axis 
Formulation  Approach.  Journal  of  Sound  and  Vibration,  1992, 156,  pp.  1-16. 

9.  Zu,  J.W.-Z.  and  Han,  R.P.S. ,  Natural  Frequencies  and  Normal  Modes  of  a  Spinning 
Timoshenko  Beam  With  General  Boundary  Conditions.  Transactions  of  the  American 
Society  of  Mechanical  Engineers,  Journal  of  Applied  Mechanics,  1992,  59,  pp.  197-204. 

10.  Katz,  R.,  Lee,  C.W.,  Ulsoy,  A.G.  and  Scott,  R.A.,  The  Dynamic  Response  of  a  Rotating 
Shaft  Subject  to  a  Moving  Load.  Journal  of  Sound  and  Vibration,  1988, 122,  pp.  131-148. 

11.  Tan,  C.A.  and  Kuang,  W.,  Vibration  of  a  Rotating  Discontinuous  Shaft  by  the  Distributed 
Transfer  Function  Method.  Journal  of  Sound  and  Vibration,  1995, 183,  pp.  451-474. 

12.  Argento,  A.  and  Scott,  R.A.,  Elastic  Wave  Propagation  in  a  Timoshenko  Beam  Spinning 
about  Its  Longitudinal  Axis.  Wave  Motion,  1995,  21,  pp.  67-74. 

13.  Dimentberg,  F.M.,  Flexural  Vibrations  of  Rotating  Shafts,  Butterworth,  London,  1961. 

14.  Dimarogonas,  A.D.  and  Paipeties,  S.A.,  Analytical  Method  in  Rotor  Dynamics,  Applied 
Science,  New  York,  1983. 

15.  Lee,  C.W.,  Vibration  Analysis  of  Rotors,  Kluwer  Academic  Publishers,  1993. 

16.  Choi,  S.H.,  Pierre,  C.  and  Ulsoy,  A.G.,  Consistent  Modeling  of  Rotating  Timoshenko  Shafts 
Subject  to  Axial  Loads.  Journal  of  Vibration  and  Acoustics,  1992, 114,  pp.  249-259. 

17.  Bhashyam,  G.R.  and  Prathap,  G.,  The  Second  Frequency  Spectrum  of  Timoshenko  Beams. 


1325 


Journal  of  Sound  and  Vibration,  1981, 76,  pp.  407-420. 

18.  Riedel,  C.H.  and  Tan,  C.  A.,  Mode  Localization  and  Delocalization  of  Constrained  Strings 
and  Beams.  Proceedings  ofASME  Biennial  Conference  on  Mechanical  Vibration  and  Noise 
(submitted),  1997 


1326 


For  simple  and  clamped  supports,  the  reflection  and  transmission  matrices  are  listed  as  follows. 


Simple  support  ( k.  =  k.  =  n 

l  =  C, 

=  c,  =  . 

Case  //  ( A  >  0 ,  B  <  0): 

1 

■  r. 

r,  ■ 

iF,  r2 

'ir, 

r./ 

'"ir.-r2 

.-ir^ 

-r^. 

CaseIV{A<0,  B<0): 

1 

-ire 

1  1 

t  “ 

'r. 

-ir: 

iT,  +r2 

.-^2 

-n_ 

i  ^T,+r2 

-r2 

•  Clamped  support  {k^  =<=<>,  m=c,  =  c,  =  J„,  =  0 );  t  =  0 . 

CaseII{A>0,  B<0): 

1  ^^1+^2  2r2 

-2iT,  -(ir,  +  r,)J’ 

CaseIV{A<0,  B<0): 

1  r-(ir,-r2)  -2iTi  ' 

^“iT.+r.L  -2r2  iT, -r^  ■ 


(32a*,  b*) 


(33a*,  b*) 


(35*) 


(36*) 


APPENDIX  II 

The  reflection  and  transmission  matrices  for  a  wave  incident  upon  a  cross-sectional  change 
for  the  simple  Euler-Bernoulli  shaft  model  can  be  determined  by  solving  the  following  sets  of 
matrix  equations.  Only  two  representative  combinations  are  shown. 

CaseII(Ai>0,  B,  <0)- Case  II  {A,>0,  B,<0): 


1327 


Casen{A,>0,  <Q)  -  Case  TV  {A,  <0 ,  <0): 


1 

1 

1 

1 1  . 

1 

1  1  . 

■  + 

- 

•^2,. 

-F^  F" 
^1/  •*■2/ 

C* 

+ 

F2  1 

rC"  = 

[T^  _r3 

L  “  1/  ^  21 

2/ J 

r,rj 

where,  F sr  and  T2r  have  been  defined  in  Eqns.  (45c,  d),  and  A^-  and  Br  are  given  by 

O’" 

G 


(39a') 

(39b') 

(46a') 

(46b*) 


If  the  rotating  shaft  is  strain-free,  then  r  can  be  reduced  to  simple  forms  representing  typical 
boundary  conditions  such  as  simple  support,  clamped  support,  and  free  end  as  shown  in  Ref.  [6]. 
Note  that  for  those  supports  in  the  strain-free  case,  the  reflection  matrices  are  constant. 


1328 


ANALYTICAL  MODELLING  OF  COUPLED  VIBRATIONS  OF 
ELASTICALLY  SUPPORTED  CHANNELS 

Yavuz  YAMAN 

Department  of  Aeronautical  Engineering,  Middle  East  Technical  University 
0653 1  Ankara,  Turkey 

An  exact  analytical  method  is  presented  for  the  analysis  of  forced  vibrations 
of  uniform  thickness,  open-section  channels  which  are  elastically  supported  at 
their  ends.  The  centroids  and  the  shear  centers  of  the  channel  cross-sections  do 
not  coincide;  hence  the  flexural  and  the  torsional  vibrations  are  coupled.  Ends 
of  the  channels  are  constrained  with  springs  which  provide  finite  transverse, 
rotational  and  torsional  stiffnesses.  During  the  analysis,  excitation  is  taken  in 
the  form  of  a  point  harmonic  force  and  the  channels  are  assumed  to  be  of  type 
Euler-Bernoulli  beam  with  St.Venant  torsion  and  torsional  warping  stiffness. 
The  study  uses  the  wave  propagation  approach  in  constructing  the  analytical 
model.  Both  uncoupled  and  double  coupling  analyses  are  performed.  Various 
response  and  mode  shape  curves  are  presented. 

1.  INTRODUCTION 

Open-section  channels  are  widely  used  in  aeronautical  structures  as  stiffeners. 
These  are  usually  made  of  beams  in  which  the  centroids  of  the  cross-section 
and  the  shear  centers  do  not  coincide.  This,  inevitably  leads  to  the  coupling  of 
possible  flexural  and  torsional  vibrations.  If  the  channels  are  symmetric  with 
respect  to  an  axis,  the  flexural  vibrations  in  one  direction  and  the  torsional 
vibrations  are  coupled.  The  flexural  vibrations  in  mutually  perpendicular 
direction  occur  independently.  In  the  context  of  this  study,  this  type  of 
coupling  is  referred  to  as  double-coupling.  If  there  is  no  cross-sectional 
symmetry,  all  the  flexural  and  torsional  vibrations  are  coupled.  This  is  called 
as  triple-coupling.  The  coupling  mechanism  alters  the  otherwise  uncoupled 
response  characteristics  of  the  structure  to  a  great  extent. 

This  problem  have  intrigued  the  scientists  for  long  time.  Gere  et  al  [1],  Lin 
[2],  Dokumaci  [3]  and  Bishop  et  al  [4]  developed  exact  analytical  models  for 
the  determination  of  coupled  vibration  characteristics.  All  those  works,  though 
pioneering  in  nature,  basically  aimed  to  determine  the  free  vibration 
characteristics  of  open-section  channels. 

The  method  proposed  by  Cremer  et  al  [5]  allowed  the  determination  of 
forced  vibration  characteristics,  provided  that  the  structure  is  uniform  in 
cross-section.  The  use  of  that  method  was  found  to  be  extremely  useful  when 
the  responses  of  uniform  structures  to  point  harmonic  forces  or  line  harmonic 
loads  were  calculated.  Mead  and  Yaman  presented  analytical  models  for  the 


1329 


analysis  of  forced  vibrations  of  Euler-Bernoulli  beams  [6].  In  that  they 
considered  finite  length  beams  ,  being  periodic  or  non-periodic,  and  studied 
the  effects  of  various  classical  or  non-classical  boundary  conditions  on  the 
flexural  response.  Yaman  in  [7]  developed  mathematical  models  for  the 
analysis  of  the  infinite  and  periodic  beams,  periodic  or  non-periodic  Kirchoff 
plates  and  three-layered,  highly  damped  sandwich  plates. 

Yaman  in  [8]  also  developed  analytical  models  for  the  coupled  vibration 
analysis  of  doubly  and  triply  coupled  channels  having  classical  end  boundary 
conditions.  In  that  the  coupled  vibration  characteristics  are  expressed  in 
terms  of  the  coupled  wave  numbers  of  the  structures.  The  structures  are  first 
assumed  to  be  infinite  in  length,  and  hence  the  displacements  due  to  external 
forcing(s)  are  formulated.  The  displacements  due  to  the  waves  reflected  from 
the  ends  of  the  finite  structure  are  also  separately  determined.  Through  the 
superposition  of  these  two,  a  displacement  field  is  proposed.  The  application 
of  the  end  boundary  conditions  gives  the  unknowns  of  the  model.  The 
analytical  method  yields  a  matrix  equation  of  unknowns  which  is  to  be  solved 
numerically.  The  order  of  the  matrix  equation  varies  depending  on  the  number 
of  coupled  waves.  If  the  cross-section  is  symmetric  with  respect  to  an  axis 
(  double-coupling)  and  if  the  warping  constraint  is  neglected,  the  order  is  six. 
If  there  is  no  cross-sectional  symmetry  (triple-coupling)  and  if  one  also 
includes  the  effects  of  warping  constraint,  the  order  then  becomes  twelve. 
This  order  is  independent  of  the  number  of  externally  applied  point  forces. 
Although  the  method  is  basically  intended  to  calculate  the  forced  response 
characteristics,  it  conveniently  allows  the  computation  of  free  vibration 
characteristics  as  well.  The  velocity  or  acceleration  of  a  point  can  easily  be 
found.  The  mode  shapes  can  also  be  determined.  Both  undamped  and  damped 
analyses  can  be  undertaken. 

This  study  is  based  on  the  models  developed  in  reference  [8]  and  aims  to 
analyze  the  effects  of  non-classical  end  boundary  conditions  on  the  coupled 
vibratory  responses.  If  the  ends  are  elastically  supported  (which  may  also  have 
inertial  properties)  the  problem  becomes  so  tedious  to  tackle  through  the 
means  of  classical  analytical  approaches.  The  current  method  alleviates  the 
difficulties  encountered  in  the  consideration  of  complex  end  boundary 
conditions. 


In  this  study  a  typical  channel,  assumed  to  be  of  type  Euler-Bernoulli  beam,  is 
analyzed.  It  represents  the  double-coupling.  Effects  of  the  elastic  end 
boundary  conditions  on  the  resonance  frequencies,  response  levels  and  mode 
shapes  are  analyzed.  Characteristics  of  otherwise  uncoupled  vibrations  are 
also  shown. 


1330 


2.  THEORY 


2.1  Flexural  Wave  Propagation  in  Uniform  Euler-Bernoulli  Beams 

Consider  a  uniform  Euler-Bernoulli  beam  of  length  L  which  is  subjected  to  a 
harmonically  varying  point  force  Fo  e  acting  at  x=Xf.  The  total  flexural 
displacement  of  the  beam  at  any  Xr  (0  <  Xr  <  L)  can  be  found  to  be  [5-8], 


w(x,,t)  =  (  i;A„e''„\  +  Fo  i  ane'^' 


r  r 


)e' 


(1) 


The  first  series  of  the  equation  represents  the  effects  of  four  waves  which  are 
being  reflected  from  the  ends  of  the  finite  beam.  They  are  called  free-waves. 
The  second  series  accounts  for  the  waves  which  are  being  created  by  the 
application  of  the  external  force  Fo  e  on  the  infinite  beam.  Those  waves  are 
known  as  forced-waves,  kn  is  the  n’th  wave  number  of  the  beam  and 
kn  =(mco^/EI)'^'‘  where  m=  Mass  per  unit  length  of  the  beam,  co=  Angular 
frequency,  EI=  Flexural  rigidity  of  the  beam,  an  values  are  the  complex 
coefficients  which  are  to  be  found  by  satisfying  the  relevant  compatibiliy  and 
continuity  conditions  at  the  point  of  application  of  the  harmonic  force  [6,7]. 
An  values,  on  the  other  hand  are  the  complex  amplitudes  of  the  free  waves  and 
are  found  by  satisfying  the  required  boundary  conditions  at  the  ends  of  the 
beam.  Once  determined,  their  substitution  to  equation  (1)  yields  the  flexural 
displacement  at  any  point  on  the  finite  beam  due  to  a  transversely  applied 
point  harmonic  force.  More  comprehensive  information  can  be  found  in  [7]. 

2.2  Torsional  Wave  Propagation  in  Uniform  Bars 

If  one  requires  to  determine  the  torsional  displacements  generated  by  a  point, 
harmonically  varying  torque,  a  similar  approach  to  the  one  given  in  Section 
2.1  can  be  used.  In  that  case,  the  total  torsional  displacement  can  be  written  as: 


=  )e‘”'  (2) 

k  is  the  wave  number  of  the  purely  torsional  wave  and  is  known  to  be 
k=(-pIoa)VGJ)^^^  .  k2=  -ki  and  GJ=Torsional  rigidity  of  the  beam,  p=Material 
density,  Io=Polar  second  moment  of  area  of  the  cross-section  with  respect  to 
the  shear  centre.  Toe'“  ^  is  the  external  harmonic  torque  applied  at  x=Xt  and 
b=l/(2kGJ).  Bn  values  are  the  complex  amplitudes  of  the  torsional  free-waves 
and  are  found  by  satisfying  the  appropriate  end  torsional  boundary  conditions. 

The  consideration  of  the  warping  constraint  To  modifies  equation  (2)  to  the 
following  form. 


1331 


(3) 


«x„t)  =  (£C„e^\  +  Toic„e-'=„'V,'  )  e 

n*l  »•! 

Now  kn  are  the  roots  of 


EFo  kn'^-GJkn^-pIoCO^=0 


(4) 


Cn  values  are  found  by  satisfying  the  necessary  equilibrium  and  compatibility 
conditions  at  the  point  of  application  of  the  point  harmonic  torque  acting  on  an 
infinite  bar  [8].  Cn  values  are  determined  from  the  end  torsional  boundary 
conditions  of  the  finite  bar. 

2.3  End  Boundary  Conditions  for  Uncoupled  Vibrations 


2.3.1  Purely  Flexural  Vibrations 

Consider  an  Euler-Bernoulli  beam  of  length  L  which  is  supported  by  springs  at 
its  ends.  The  springs  provide  finite  transverse  and  rotational  constraints  Kt  and 
Kr  respectively.  The  elastic  end  boundary  conditions  can  be  foimd  to  be: 

El  w”(0)  -  Kr  I  w’(0)=0  El  w’”(0)  +  Ktj  w(0)=0 

El  w”(L)  +  K  r,r  w’(L)=0  El  w’”(L)  -  Kt,rW(L)=0  (5) 


Here  w’= 


dw(x) 


w 


d‘w(x) 


and  w’” 


dV(x) 


w(x)  is  the  spatially 


dx  ’  dx“  dx' 

dependent  part  of  equation  (1)  and  second  subscripts  1  and  r  allows  one  to  use 
different  stiffnesses  for  left  and  right  ends.  A  more  comprehensive  study  on 
these  aspects  can  be  found  in  references  [6,7]. 


2.3.2  Purely  Torsional  Vibrations 

Now  consider  a  bar  of  length  L  which  is  supported  by  torsional  springs, 
having  finite  Ktor,at  its  ends.  The  elastic  end  boundary  conditions  requires  that, 

Torque  (0)  -  Ktor,i  (|)(0)=0  and  Torque  (L)  +  Ktor  .r  (i)(L)=0  (6) 

Depending  on  the  consideration  of  the  warping  constraint  To,  the  torque  has 
the  following  forms 

Torque(x)=GJ  — —  or  Torque(x)=  GJ  — —  -  Ei  o  ,  3  ( /) 

dx  dx  ux 


1332 


(})(x)  in  equation  (7)  should  be  obtained  either  from  equation  (2)  or  equation  (3) 
depending  on  the  warping  constraint  r©. 

2.4  Doubly-Coupled  Vibrations 

Now,  consider  Figure  1  .  It  defines  a  typical  open  cross-section  which  is 
synunetric  with  respect  to  y  axis 


V  z 


(b) 

Figure  1  :  A  Typical  Cross-section  of  Double-coupling 

(  a.  Coordinate  System,  b.  Real  and  Effective  Loadings 
C:  Centroid,  O:  Shear  Centre) 

A  transverse  load  applied  through  C  results  in  a  transverse  load  through  O  and 
a  twisting  torque  about  O.  In  this  case  the  flexural  vibrations  in  z  direction  are 
coupled  with  the  torsional  vibrations  whereas  the  flexural  vibrations  in  y 
direction  occur  independently.  The  motion  equation  of  the  coupled  vibrations 
is  known  to  be  [1,2]. 


a-w  a-(j) 


=  0 


a‘‘(i)  3^6  a’w  a"(j) 


(8) 


If  one  assumes  that, 


w(x,t)  =  w„eVe”' 

(|)(x.t)  =  4>„eVe“’  (9) 


1333 


Then,  it  can  be  found  that,  a  load  Pz  through  the  centroid  will  create  the 
following  displacements  at  any  x  (0  <  x  <  L)  along  the  length  of  the 
channel  [8], 

w(x,t)  =  (2A„e^’‘+Pjt  a„  e'^  )e'"' 

n^l  nvl 

n«l  n>I 

Now  kn  values  are  the  coupled  wave  numbers,  An  values  are  the  complex 
amplitudes  of  the  coupled  free  waves,  an  values  are  the  complex  coefficients 
which  are  to  be  found  by  satisfying  the  required  compatibility  and  continuity 
conditions  and  'Pn^  (  (El^  kn'^-mco^)  /  (CymcD^) )  [8]. 

If  required,  the  warping  displacement  u(x,t)  can  be  found  from  (l)(x,t  )as 

u(x,t)=-2A,^iM  (11) 

dx 

where  As  is  the  swept  area. 

Here  2j  gives  the  order  of  the  motion  equation.  j=3  defines  the  case  in  which 
the  effects  of  warping  constraint  are  neglected  and  j=4  represents  the  case 
which  includes  the  warping  effects. 


An  values  are  found  by  satisfying  the  necessary  2j  end  boundary  conditions.  If 
warping  constraint  is  neglected,  the  required  six  boundary  conditions  have  the 
general  forms  given  in  equations  (5)  and  (6).  But  the  forms  of  w(x)  and  (j)(x) 
are  now  those  given  by  equations  (10)  with  j=3.  If  the  warping  constraint  is 
included  in  the  analysis  the  boundary  conditions  become  eight.  The  six  of 
those  are  again  found  by  considering  equations  (10)  with  j=4  and  substituting 
the  resultant  forms  into  equations  (5)  and  (6).  The  remaining  two  can  be 
found  by  evaluating  equation  (1 1)  at  both  ends. 

When  the  flexural  and  torsional  displacement  expressions  are  substituted  into 
the  relevant  equations,  a  set  of  equations  is  obtained.  For  the  case  of  a  load 
Pz  and  no  warping  constraint,  the  following  equations  can  be  found  for  j=3. 

EI^  w’”(0)  +  Kt,iw(0)=0 

EI^  (E  kn^  An+  (-1)  PzZ  -  kn^  a  „  6  '  "f  '  ) 

^t,l  (  E  ■^n  ■^PzE^n^  n  f  )  “0  (12) 


1334 


(13) 


EI^w”(0)  -Kr,i  w’(0)=0: 

EI^(  I  kn'An  +  Pz  t  kn'an  6 ‘  '  ) 

n«l  n«t 

■  (  S  kn  An  +  Pz^'kn^n®  n  f  ) 

n*l  n“l 


GJ^^l«=o-K,„,.i  W)=0: 

dx 

GJ(  S  k„  T „  A„  +  (- 1 )  (Pz  cy)  t  -k„  'f  „  a„  e  '  ’‘f ') 

fl«l  11' I 

-K,or,i  (  i  %  A„  +  (Pz  Cy)  i  >?„  a„  e  ')=0 


diKx)  I 

GJ-^Ix.l  +  K,„„  <KL)=0: 
dx 

GJ(  2k„>P„  A„  e“+  (PzCy)  i-k„'P„a„  ) 

ns  I  11=1 

+  K,„,.r  ( I  %  A„  +  (Pz  Cy)  t  Tn  a„  e  "f  ')=0 


El5W”(L)+  Kr,rW’(L)=0: 

El^CS  k„^A„e^‘-+(Pz)  i  k„^a„  ) 

n-l  ii’*) 

+  Kz,r(  i  k„A„  e“+  (Pz)  t-k„a„  e■^'‘-^')=0 

n»l  n=i 


(14) 


(15) 


(16) 


EI^  w’”(L)-Kt,rW(L)=0: 

Eiaz  kn'Ane'n^  +(Pz)  2  -  kn^  a  „  e n  ' ^  ) 

11=1  Iia] 

-K,|(Z  A„eV+(Pz)  Za„e^''-V)=0  (17) 

lt=l  11=1 

Here  (-1)  multipliers  are  included  due  to  the  symmetry  and  anti-symmetry 
effects. 

Those  equations  can  be  cast  into  the  following  matrix  form. 


=  -{Terms  containing  Pz}  (18) 


1335 


An  eighth  order  equation  represents  the  necessary  matrix  equation  for  the 
determination  of  An  values  if  the  warping  constraint  is  included  in  the  analysis. 
In  that  case,  equations  (12),  (13),  (16)  and  (17)  are  valid  with  j=4.  On  the  other 
hand  equations  (14)  and  (15)  should  be  replaced  by, 

(GJ^-Er„^^)L.o  <l>(0)=0: 

dx  dx 

(GJ(  £  k„  %  A„  +  (- 1 )  (P.  Cy)  £  -k„  a„  e  *  „ '  ’‘f ')  - 

nol  n»l 

Er„(  £  k„3  'f  „  A„  +  (- 1 )  (P^  Cy)  £  -k„  =  %  a„  e  *  „ '  -f ')) 

it«l  IIS’! 

-K,„,,,(|;'P„A„+(P,Cy)i  'P„a„e-^'’‘f')=0  (19) 

cr  A/T^=A. 

(GJ  “  EFo  j  3  )•  x=L  I^tor>r  y(L)  0  . 
dx  dx 

(GJ(  Jkn'Fn  A„  e’=„‘-+  (PjCy)  i-kn'Pnan  '  )- 

n=t  Ii»l 

EFoC  Xkn^  'J'n  A„  eV+  (PzCy)  £-k„^'P„a„  ' )) 

11=11  n-l 

+  Kior,,  (  X  %  A„  +  (Pz  Cy)  X  %  a„  e  *  „ '  \  ')=0  (20) 

n-]  ii3| 

where  j=4.  The  remaining  two  equations  are  found  by  considering  the  warping 
of  the  extreme  ends.  If  the  ends  are  free  to  warp  the  axial  stress  is  zero,  if  the 
ends  are  not  to  warp  the  axial  displacements  are  zero  at  both  ends.  No  elastic 
constraints  are  imposed  on  end  warping.  If  the  left  end  is  free  to  warp  and  the 
right  end  is  not  to  warp,  the  required  boundary  conditions  can  be  shown  to  be; 

u’(0)=0:(x  kn^'Pn  A„  +  PzX  'i'.ian  k„^e-^  '  ^')  =  0  (21) 

u(L)=0  :(X  k„  e^‘'  %  A„  +  PzX  (-k„)  '‘-"f')  =  0  (22) 

n=]  list 

All  the  equations  can  be  put  into  the  following  matrix  form 

A.' 

A: 

A, 

•  =  -{Terms  containing  Pz}  (23) 

A, 

A. 

A. 

A. 


1336 


Required  An  values  are  numerically  found  from  equations  (18)  or  (23).  Their 
substitution  to  the  appropriate  forms  of  equations  (10)  and  (11)  yield  the 
required  responses  at  any  point  on  the  beam. 

3.RESULTS  AND  DISCUSSION 

The  theoretical  model  used  in  the  study  is  shown  in  Figure  1  and  has  the 
following  geometric  and  material  properties: 

L=l(m),  A  =1.0*10"' (m\  h  =  5.0*10-^(m),  15  =  4.17*10-*  (m\ 

Cy  =  15.625*10'*  (m),  J  =  3.33* lO"' '  (m*),  =7.26*10'*  (m"'),  p=  2700  (kg/m*), 
r„  =2.85*10''*  (m*),  E  =  7*10'“  (N/m*),  G  =  2.6*  10'“  (N/m*). 

Structural  damping  for  torsional  vibrations  is  included  through,  complex 
torsional  rigidity  as  GJ*=GJ(l+z|3).  For  coupled  vibrations,  it  is  also  included 
through  the  complex  flexural  rigidity  as  El^(H-z'n). 

First  presented  are  the  results  for  purely  torsional  vibrations.  A  bar  assumed  to 
have  the  given  L,  p,  lo,  G  and  J  values  is  considered.  The  bar  is  then  restrained 
at  both  ends  by  springs  having  the  same  torsional  stiffness  Ktor-  A  very  low 
damping,  p=10'^,  is  assigned  and  the  resonance  frequencies  are  precisely 
determined.  It  is  found  that,  the  introduction  of  a  small  Ktor  introduces  a  very 
low  valued  resonance  frequency.  That  fundamental  frequency  increases  with 
increasing  Ktor  and  as  torsional  constraint  reaches  to  very  high  values,  it 
approaches  to  the  fundamental  natural  frequency  of  torsionally  fixed-fixed 
beam.  Table  1  gives  the  fundamental  frequencies  for  a  range  of  Ktor  values  and 
Figure  2  represents  the  fundamental  mode  shapes  for  selected  Ktor  values. 


Table  1:  Uncoupled  Fundamental  Torsional  Resonance  Frequencies 

((3=10’^,No  Warping  Constraint) 


Ktor  [N1 


Frequency  [Hz] 


0 

0. 

10'^ 

1.606 

10'' 

5.035 

io“ 

14.683 

10' 

28.407 

10^ 

32.678 

10^ 

33.187 

10^ 

33.239 

;10*“ 

33.245 

1337 


Figure  6  on  the  other  hand  represents  the  low  frequency  torsional  receptances 
of  the  case  in  which  the  warping  constraint  is  taken  into  consideration  and 
the  ends  are  free  to  warp.  This  graph  is  included  in  order  to  show  the  variation 
of  fundamental  torsional  resonance  frequencies  for  a  range  of  Ktor  values. 


_ -  K,3r= 

K„,= 

kK,„= 


5*10': 

1*10': 

5-10" 

1-10' 

5-10' 

1*10® 


Figure  6.  Fundamental  Frequencies  of  Purely  Torsional  Vibrations 
((3=0.01,  x=0.13579[m], Warping  Constraint  Included,  Ends  are  Free  to  Warp) 


The  second  part  of  the  study  investigates  the  characteristics  of  doubly-coupled 
vibrations.  Now,  the  effects  of  each  constraint  are  separately  considered.  A 
channel  having  the  given  parameters  is  supported  at  its  ends  by  springs  Kt,  Kr, 
and  Ktor-  Warping  constraint  is  included  in  the  analysis  and  the  ends  are 
assumed  to  warp  freely.  First  analyzed  is  the  effects  of  Ktor-  For  this  Kt=10^° 
[N/m]  and  Kr  =10^°  [N]  are  assigned  at  both  ends  of  the  channel  and  kept  fixed 
throughout  the  study.  Ktor  is  varied  and  the  frequencies  are  shown  in  Table  2. 


Table  2.  Effects  of  Ktnr  in 

Doublv-counled  Vibrations 

(Kt=10^°  [N/m]  and  Kr=10^®  [N],  Warping  Constraint  Included) 

A:  First  Torsion  Dominated  Frequency  [Hz] 

B:  First  Flexure  Dominated  Frequency  [Hz] 

Ktor  [Nl 

A 

B 

10-^ 

1.607 

134.603 

10'' 

5.069 

134.936 

10° 

15.620 

138.258 

10' 

39.725 

169.247 

10^ 

56.691 

318.423 

10^ 

59.528 

422.061 

10^ 

59.828 

430.642 

1338 


0.0  0.2  0.4  0.6  0.8  1.0 


ND  LENGTH 

Figure  2.  Fundamental  Mode  Shapes  of  Purely  Torsional  Vibrations 
((3=0,  No  Warping  Constraint) 

Then,  the  warping  constraint  To  is  included  in  the  analysis  and  the  results  of 
purely  torsional  vibrations  are  presented  again.  The  beam  had  the  same  Ktor 
values  at  both  ends  and  the  numerical  values  of  the  relevant  parameters  are 
taken  to  be  those  previously  defined.  Figure  3  represents  the  fundamental 
mode  shapes  for  which  the  ends  are  free  to  warp,  whereas  Figure  4  shows  the 
mode  shapes  of  the  case  in  which  there  is  no  warping  at  the  ends. 


Figure  3.  Fundamental  Mode  Shapes  of  Purely  Torsional  Vibrations 


(P=0,  Warping  Constraint  Included,  Ends  are  Free  to  Warp) 


(P=0,  Warping  Constraint  Included,  Ends  Can  Not  Warp) 


Figure  5  is  drawn  to  highlight  the  effects  of  end  warping.  Both  ends  of  the 
channel  are  restrained  with  Ktor=l  *  lO’  [N]  and  all  the  other  parameters  of  the 
study  are  kept  fixed.  Figure  5  represents  the  direct  torsional  receptances  of 
two  cases  in  which  the  ends  of  the  channel  are  allowed  to  warp  and  not  to 
warp  in  turn.  It  can  be  seen  that  the  prevention  of  end  warping  increases  the 
resonant  frequencies. 


Figure  5.  Frequency  Response  of  Purely  Torsional  Vibrations 

([3=0.01,  X  =0.13579  [m],  Ktor=l*10'  [N]»  Warping  Constraint  Included) 


1340 


It  can  be  seen  that,  when  it  has  lower  values  Ktor  is  more  effective  on  the 
torsion  dominated  resonance  frequencies.  For  the  higher  Ktor  values,  the 
effects  are  more  apparent  on  the  flexure  dominated  frequencies. 

Figure  7  represents  the  direct  flexural  receptance  of  the  channel  for 
a  set  of  selected  end  stiffnesses.  Torsion  dominated  resonances  at  59.528  [Hz], 
206.071  [Hz]  and  476.649  [Hz]  appear  as  spikes.  The  flexure  dominated 
resonance  occurs  at  422.061  Hz. 


Figure  7.  Frequency  Response  of  Doubly-coupled  Vibrations 
(ti=0.001,  P=0.001,  x=0.13579[m],  Warping  Constraint  Included, 

Ends  are  free  to  warp,  Kr=1.10^°[N],  Kt=1.10^°  [N/m],  Ktor=l-10^  [N] ) 

Then  the  effects  of  the  rotational  spring,  Kr,  are  considered.  The  ends  of  the 
channel  are  assumed  to  be  restrained  with  Kt=10^°  [N/m]  and  Ktor  =10^°  [N]. 
The  resulting  frequencies  are  given  in  Table  3  for  a  range  of  Kr  values. 


Table  3.  Effects  of  Kr  in  Doublv-coupled  Vibrations 

(Kt=10^°  [N/m]  and  Ktor=10^°  [N],  Warping  Constraint  Included) 
A;  First  Torsion  Dominated  Frequency  [Hz] 

B:  First  Flexure  Dominated  Frequency  [Hz] _ 


Kr[N] _ A _  B 


10*^ 

58.678 

205.344 

10° 

58.679 

205.358 

10^ 

58.699 

206.688 

10^ 

58.858 

218.055 

10" 

59.437 

289.738 

10^ 

59.799 

405.667 

10^ 

59.855 

429.345 

1341 


It  is  seen  that  Kr  is  not  effective  on  torsion  dominated  resonance  frequencies, 
but  plays  significant  role  for  flexure  dominated  resonance  frequencies. 

Finally  considered  the  effects  of  the  transverse  spring  Kt.  Again,  the  channel  is 
assumed  to  have  very  high  Kr  and  Ktor  values  at  both  ends  and  Kt  values  are 
varied.  Table  4  shows  the  resonance  frequencies. 


Table  4.  Effects  of  Kt  in  Doublv-coupled  Vibrations 


(Kr=10^°  [N]  and  Ktor =10^°  [N],  Warping  Constraint  Included) 
A:  First  Torsion  Dominated  Frequency  [Hz] 

B:  First  Flexure  Dominated  Frequency  [Hz] _ 


Kt  [N/m] 

A 

B 

10' 

1.369 

70.455 

10^ 

4.328 

70.501 

10^ 

13.592 

70.973 

10^ 

39.282 

77.379 

10= 

58.095 

159.665 

10= 

59.699 

361.255 

10* 

59.860 

431.215 

It  can  be  seen  that  the  transverse  stiffness,  like  torsional  stiffness,  effects  both 
flexure  and  torsion  dominated  frequencies. 


4.CONCLUSIONS 


In  this  study,  a  new  analytical  method  is  presented  for  the  analysis  of  forced 
vibrations  of  open  section  channels  in  which  the  flexible  supports  provide  the 
end  constraints.  The  dynamic  response  of  open  section  channels  is  a  coupled 
problem  and  their  analysis  requires  the  simultaneous  consideration  of  all  the 
possible  vibratory  motions.  The  wave  propagation  approach  is  an  efficient  tool 
for  this  complicated  problem  and  the  developed  method  is  based  on  that. 

The  current  method  analyzes  the  forced,  coupled  vibrations  of  open  section 
channels.  The  channels,  taken  as  Euler-Bernoulli  beams,  have  uniform  cross- 
section  and  a  single  symmetry  axis.  That  consecutively  leads  to  the  coupling 
of  flexural  vibrations  in  one  direction  and  torsional  vibrations.  The  excitation 
is  assumed  to  be  in  the  form  of  a  harmonic  point  force,  acting  at  the  centroid. 


1342 


Various  frequency  response  curves  of  uncoupled  and  coupled  vibrations  are 
presented  for  a  variety  of  different  elastic  end  boundary  conditions(  which  may 
also  have  the  inertial  properties).  The  developed  method,  although  aimed  at 
determining  the  forced  vibration  characteristics,  is  also  capable  of  determining 
the  free  vibration  properties.  This  is  also  demonstrated  by  presenting  various 
mode  shape  graphs.  It  has  been  determined  that  the  transverse  and  the  torsional 
stiffnesses  play  more  significant  role  as  compared  to  the  rotational  stifness. 

The  method  can  be  used  in  analyzing  the  effects  of  multi  point  and/or 
distributed  loadings.  This  can  simply  be  achieved  by  modifying  the  terms  of 
the  forcing  vector  without  increasing  the  order  of  the  relevant  matrix  equation. 
The  developed  method  can  also  be  used  in  the  analysis  of  elastically 
supported,  triply-coupled  vibrations  of  uniform  channels.  Results  of  that  study 
will  be  the  subject  of  another  paper. 


REFERENCES 

1.  Gere,  J.M.  and  Lin,  Y.K.,  Coupled  Vibrations  of  Thin-Walled  Beams  of 
Open  Cross-Section.  J.  Applied Mech  Trans. ASME.,\9SZ,  80,373-8. 

2.  Lin,  Y.K.,  Coupled  Vibrations  of  Restrained  Thin-Walled  Beams. 

J.  Applied  Mech.  Trans. ASME.,  1960,  82,  739-40. 

3.  Dokumaci,  E.,  An  Exact  Solution  for  Coupled  Bending  and  Torsional 
Vibrations  of  Uniform  Beams  Having  Single  Cross-Sectional  Symmetry. 
JSoundandVib.Am,  119,443-9. 

4.  Bishop,  R.E.D,  Cannon,  S.M.  and  Miao,  S.,  On  Coupled  Bending  and 
Torsional  Vibration  of  Uniform  Beams.  J.Sound  and  Fi'/)., 1989,131,457-64. 

5.  Cremer,  L.  and  Heckl,  y\..,Structure~  Borne  Sound,  Springer-Verlag,1988. 

6.  Mead,  D.J.  and  Yaman,  Y.,  The  Harmonic  Response  of  Uniform  Beams  on 
Multiple  Linear  Supports:  A  Flexural  Wave  Analysis.  J.  Sound  and  Vib, 
1990,  141,465-84 

7.  Yaman,  Y.  Wave  Receptance  Analysis  of  Vibrating  Beams  and  Stiffened 
Plates.  PA  Z).  Ttew,  University  of  Southampton,  1989. 

8  Yaman,  Y.,  Vibrations  of  Open-Section  Channels:  A  Coupled  Flexural  and 
Torsional  Wave  Analysis.  (J.  Sound  and  Vib,  Accepted  for  publication) 


1343 


1344 


THE  RESPONSE  OF  TWO-DIMENSIONAL  PERIODIC  STRUCTURES 
TO  HARMONIC  AND  IMPULSIVE  POINT  LOADING 


R.S.  Langley 

Department  of  Aeronautics  and  Astronautics 
University  of  Southampton 
Southampton  S017  IBJ 


ABSTRACT 

Much  previous  work  has  appeared  on  the  response  of  a  two-dimensional 
periodic  structure  to  distributed  loading,  such  as  that  arising  from  a  harmonic 
pressure  wave.  In  contrast  the  present  work  is  concerned  with  the  response 
of  a  periodic  structure  to  localised  forcing,  and  specifically  the  response  of  the 
system  to  both  harmonic  and  impulsive  point  loading  is  considered  by 
employing  the  method  of  stationary  phase.  It  is  shown  that  the  response  can 
display  a  complex  spatial  pattern  which  could  potentially  be  exploited  to 
reduce  the  level  of  vibration  transmitted  to  sensitive  equipment. 

1.  INTRODUCTION 

Many  types  of  engineering  structure  are  of  a  repetitive  or  periodic 
construction,  in  the  sense  that  the  basic  design  consists  of  a  structural  unit 
which  is  repeated  in  a  regular  pattern,  at  least  over  certain  regions  of  the 
structure.  An  orthogonally  stiffened  plate  or  shell  forms  one  example  of  an 
ideal  two-dimensional  periodic  structure  in  which  the  fundamental  structural 
unit  is  an  edge  stiffened  panel.  Although  a  completely  periodic  structure  is 
unlikely  to  occur  in  practice,  much  can  be  ascertained  regarding  the  structural 
dynamic  properties  of  a  real  structure  by  considering  the  behaviour  of  a 
suitable  periodic  idealization.  For  this  reason,  much  previous  work  has  been 
performed  on  the  dynamic  behaviour  of  two-dimensional  periodic  structures, 
with  particular  emphasis  on  free  vibration  and  the  response  to  pressure  wave 
excitation  [1,2].  However,  no  results  have  yet  been  appeared  regarding  the 
response  of  two-dimensional  periodic  structures  to  point  loading  (as  might 
arise  from  equipment  mounts),  and  this  topic  forms  the  subject  of  the  present 
work.  A  general  method  of  computing  the  response  to  both  harmonic  and 
impulsive  loading  is  presented,  and  this  is  then  applied  to  an  example  system. 

Initially  the  response  of  a  two-dimensional  periodic  structure  to  harmonic 
point  loading  is  considered,  and  it  is  shown  that  the  far-field  response  can  be 
expressed  very  simply  in  terms  of  the  "phase  constant"  surfaces  which 
describe  the  propagation  of  plane  waves.  It  is  further  shown  that  for 


1345 


excitation  within  a  pass  band  two  distinct  forms  of  response  can  occur;  in  the 
first  case  the  amplitude  of  the  response  has  a  fairly  smooth  spatial  distribution, 
whereas  in  the  second  case  a  very  uneven  distribution  is  obtained  and  "shadow 
zones"  of  very  low  response  are  obtained.  The  second  form  of  behaviour  is 
related  to  the  occurrence  of  caustics  (defined  in  section  3.3),  and  the 
distinctive  nature  of  the  response  suggests  that  a  periodic  structure  might  be 
designed  to  act  as  a  spatial  filter  to  isolate  sensitive  equipment  from  an 
excitation  source. 

Attention  is  then  turned  to  the  impulse  response  of  a  two-dimensional  periodic 
structure.  It  is  again  shown  that  the  response  can  be  expressed  in  terms  of  the 
phase  constant  surfaces  which  describe  the  propagation  of  plane  waves.  The 
application  of  the  method  of  stationary  phase  to  this  problem  has  a  number  of 
interesting  features,  the  most  notable  being  the  fact  that  four  or  more 
stationary  points  can  arise.  It  is  found  that  a  surface  plot  of  the  maximum 
response  amplitude  against  spatial  position  reveals  features  which  resemble  the 
"caustic"  distributions  obtained  under  harmonic  loading. 

2.  RESPONSE  TO  A  HARMONIC  POINT  LOAD 


2.1  Modal  Formulation  and  Extension  to  the  Infinite  System 
A  two-dimensional  periodic  structure  consists  of  a  basic  unit  which  is  repeated 
in  two  directions  to  form  a  regular  pattern,  as  shown  schematically  in  Figure 
1.  Each  unit  shown  in  this  figure  might  represent  for  example  an  edge 
stiffened  curved  panel  in  an  aircraft  fuselage  structure,  a  three-dimensional 
beam  assembly  in  a  roof  truss  structure,  or  a  pair  of  strings  in  the  form  of  a 
"-f"  in  a  cable  net  structure.  The  displacement  w  of  the  system  can  be 
written  in  the  form  w{n,x),  where  n={n^  nf)  identifies  a  particular  unit  and 
x  =  {Xi  X2  X3)  identifies  a  particular  point  within  the  unit.  The  coordinate 
system  x  is  taken  to  be  local  to  each  unit,  and  the  precise  dimension  of  both 
X  and  the  response  vector  w  will  depend  on  the  details  of  the  system  under 
consideration. 

The  present  section  is  concerned  with  the  response  of  a  two- 
dimensional  periodic  structure  to  harmonic  point  loading  of  frequency  co.  In 
the  case  of  a  system  of  finite  dimension,  the  response  at  location  (n,x)  to  a 
harmonic  force  F  applied  at  (0,jCo)  can  be  expressed  in  the  standard  form  [3] 


»’(n,*)=EE 

P  9 


(1) 


where  rj  is  the  loss  factor,  4>p^(n,x)  are  the  modes  of  vibration  of  the  system 
and  (j}pg  are  the  associated  natural  frequencies.  The  modes  ^p^  which  appear 
in  equation  (1)  are  scaled  to  unit  generalized  mass,  so  that 


1346 


/!,  «!  V 


(2) 


where  V  represents  the  volume  (or  equivalent)  of  a  unit  and  p(jc)  is  the  mass 
density.  The  present  concern  is  with  the  response  of  an  infinite  system,  or 
equivalently  the  response  of  a  large  finite  system  in  which  the  vibration  decays 
to  a  negligible  level  before  reaching  the  system  boundaries.  In  this  case  the 
response  is  independent  of  the  system  boundary  conditions,  and  it  follows  that 
any  analytically  convenient  set  of  modes  can  be  employed  in  equation  (1).  As 
explained  in  reference  [4],  it  is  expedient  to  consider  the  Born- Von  Karman 
(or  "periodic”)  boundary  conditions,  as  in  this  case  the  modes  of  vibration  can 
be  expressed  very  simply  in  terms  of  propagating  plane  wave  components. 
In  this  regard  it  can  be  noted  from  periodic  structure  theory  [5]  that  a 
propagating  plane  wave  of  frequency  w  has  the  general  form 

w(rt,jc)=i?^{^(jc)exp(zej«,  +ie^n^nu)t)} ,  (3) 


where  and  eo  are  known  as  the  propagation  constants  of  the  wave  (with  - 
T<e,<T  and  -7r<G2<T  for  uniqueness),  and  g{x)  is  a  complex  amplitude 
function.  By  considering  the  dynamics  of  a  single  unit  of  the  system  and 
applying  Bloch’s  Theorem  [5],  it  is  possible  to  derive  a  dispersion  equation 
which  must  be  satisfied  by  the  triad  (w, 61,62)  -  by  specifying  Gj  and  €2  this 
equation  can  be  solved  to  yield  the  admissible  propagation  frequencies  w.  By 
way  of  example,  solutions  yielded  by  this  procedure  for  a  plate  which  rests  on 
a  grillage  of  simple  supports  are  shown  in  Figure  2  (after  reference  [6]).  It 
is  clear  that  the  solutions  form  surfaces  above  the  61-62  plane  -  these  surfaces 
are  usually  referred  to  as  "phase  constant"  surfaces,  and  a  single  surface  will 
be  represented  here  by  the  equation  a;= 0(61,62).  The  phase  constant  surfaces 
always  have  cyclic  symmetry  of  order  two,  so  that  0(ei,62)=0(-6i,-62);  for  an 
orthotropic  system  the  surfaces  also  have  cyclic  symmetry  of  order  four,  and 
therefore  only  the  first  quadrant  of  the  61-62  plane  need  be  considered 
explicitly,  as  in  Figure  2. 

The  key  point  about  the  Born- Von  Karman  boundary  conditions  is  that 
a  single  propagating  wave  can  fully  satisfy  these  conditions  providing  and 
6o  are  chosen  appropriately.  The  conditions  state  that  the  left  hand  edge  of  the 
system  is  contiguous  with  the  right  hand  edge,  and  similarly  the  top  edge  is 
contiguous  with  the  bottom  edge,  so  that  the  system  behaves  as  if  it  were 
topologically  equivalent  to  a  torus.  If  the  system  is  comprised  of  XN2 
units,  then  a  propagating  wave  will  satisfy  these  conditions  if  and 

62^2 =2x^  for  any  integers  p  and  q.  Following  equation  (3),  the  displacement 
associated  with  such  a  wave  can  be  written  in  the  form 


1347 


(4) 


where  ei^  and  €2,  are  the  appropriate  values  of  the  phase  constants,  and 
<^^^=0(ej^,e2g)-  Now  since  it  follows  that  a  wave  of 

frequency  travelling  in  the  opposite  direction  to  will  also  satisfy  the 
boundary  conditions.  This  wave  say)  will  have  the  form 

where  it  has  been  noted  from  periodic  structure  theory  that  reversing  the 
direction  of  a  wave  leads  to  the  conjugate  of  the  complex  amplitude  function 
^(x).  The  two  waves  represented  by  equations  (4)  and  (5)  can  be  combined 
with  the  appropriate  phase  to  produce  two  modes  of  vibration  of  the  system 
in  the  form 


KSn,x) 

• 

Re 

Im 

(6) 


By  adopting  this  set  of  modes  it  can  be  shown  [4]  that  equation  (1)  can  be  re¬ 
expressed  as 

K.(«,x)=  £  £  2g;WF^g„(xJexp(-ie,^«,-%,n,) 

^=1-^/2  9=1  -Nj/2  +Z?7)  -(J? 


where  and  M  have  been  taken  to  be  even,  and  the  amplitude  function  gp^ 
is  scaled  so  that 

g^,=[2p(x)WV,Ar,]-%(x),  {llV)\^f„(x)f;,(x)dx=\,  (8,9) 

where  the  normalized  amplitude  function  fp^  is  defined  accordingly.  The 
summation  which  appears  in  equation  (7)  includes  only  those  modes  associated 
with  a  single  phase  constant  surface  0(€i,62);  if  more  than  one  surface  occurs 
then  the  equation  should  be  summed  over  the  complete  set  of  surfaces.  The 
summation  will  include  modes  for  each  surface,  which  is  consistent  with 
known  results  for  the  modal  density  of  a  two-dimensional  periodic  structure. 

Equation  (7)  yields  the  response  of  a  finite  system  of  dimension  XM 
to  a  harmonic  point  load  -  this  response  is  identical  to  that  of  an  infinite 
system  if  the  vibration  decays  to  a  negligible  amount  before  meeting  the 
system  boundaries.  If  the  system  size  is  allowed  to  tend  to  infinity  in  equation 


1348 


(7)  then  neighbouring  values  of  the  phase  constants  e^p  and  €2^  become  closely 
spaced  (since  deip=ei_p+i-eip=2T/A^i  and  de2g= €2,9+ 2x77/2) >  and  in  this 
case  the  summations  can  be  replaced  by  integrals  over  the  phase  constants  to 
yield 


-  LI  me„e,)ninv)-c^^ 


where  0)= 0(61,62)  and  ^(x)  is  the  complex  amplitude  associated  with  the  wave 
(£0,61,62).  The  evaluation  of  the  integrals  which  appear  in  equation  (10)  is 
discussed  in  the  following  sub-sections. 


2.2  Integration  over  61 

The  integral  over  6i  which  appears  in  equation  (10)  can  be  evaluated  by  using 
contour  integration  techniques.  Two  possible  contours  in  the  complex  e,  plane 
are  shown  in  Figure  3;  to  ensure  a  zero  contribution  from  the  segment 
Im(6i)  =  ±oo,  the  upper  contour  is  appropriate  for  /Zi<0  while  the  lower 
contour  should  be  used  for  n^X).  For  each  contour  the  contributions  from 
the  segments  and  61;;= x  cancel,  since  the  integrand  which  appears  in 

equation  (10)  is  unchanged  by  an  increment  of  2x  in  the  real  part  of  ei.  The 
only  non-zero  contribution  to  the  integral  around  either  contour  therefore 
arises  from  the  segment  which  lies  along  the  real  axis.  The  poles  of  the 
integrand  occur  at  the  61  solutions  of  the  equation 

[fl(6i,62)?(U/i7)-a;^=0,  (11) 

for  specified  62  and  oj.  By  definition  there  will  be  two  real  solutions^  in  the 
absence  of  damping  (77=0)  providing  the  frequency  range  covered  by  the 
phase  constant  surface  includes  oj.  Any  complex  solutions  to  equation  (11)  in 
the  absence  of  damping  will  correspond  physically  to  "evanescent"  waves 
which  decay  rapidly  away  from  the  applied  load.  The  present  analysis  is 
concerned  primarily  with  the  response  of  the  system  in  the  far  field  (that  is, 
at  points  remote  from  the  excitation  source),  and  for  this  reason  attention  is 
focused  solely  on  those  roots  to  equation  (11)  which  are  real  when  77=0.  The 
effect  of  damping  on  these  roots  can  readily  be  deduced:  if  77  is  small  then  it 
follows  from  equation  (11)  that  a  real  solution  6^  will  be  modified  to  become 
6I-i(o7/2)(5Q/^6l)■^  and  hence  the  real  pole  for  which  dn/36i<0  is  moved  to 
the  upper  half  plane,  while  that  for  which  30/56,  >0  is  moved  to  the  lower 
half  plane.  Given  that  the  residue  at  such  a  pole  is  proportional  to  (30/36i)'\ 


^One  positive  and  one  negative.  These  solutions  will  have  the  form  ±6, 
for  an  orthotropic  system. 


1349 


it  follows  that  the  sign  of  the  residue  which  arises  from  the  contour  integral 
is  determined  by  the  integration  path  selected,  and  hence  by  the  sign  of 
These  considerations  lead  to  the  result 


~Tr~l  fi|3fi/3«,|(l+/)/) 


(12) 


where  e,(£2,M)  is  the  appropriate  solution  to  equation  (11).  The  evaluation  of 
the  integral  over  €3  is  discussed  in  the  following  section. 

3.3  Integration  over  €3 

Since  the  present  concern  is  with  the  response  of  the  system  at  some  distance 
from  the  excitation  point,  the  integral  over  e,  which  appears  in  equation  (12) 
can  be  evaluated  to  an  acceptable  degree  of  accuracy  by  using  the  method  of 
steepest  descent  [7],  With  this  approach  it  is  first  necessary  to  identify  the 
value  of  €2  for  which  the  exponent  -i(eirti+e2«2)  is  stationary.  The  condition 
for  this  is 

{3ejde^n^+n^=0.  (13) 


Now  Gi  and  satisfy  the  dispersion  relation,  equation  (11),  and  thus  equation 
(13)  can  be  re-expressed  in  the  form 

(aQ/a62)«r(5^2/a€,)«2=o,  (i4) 


where  it  has  been  noted  from  equation  (11)  that,  for  fixed  co,  3ei/3€2=- 
(5Q/3e2)/(9fi/36i),  In  the  absence  of  damping  the  wave  group  velocity  lies  in 
the  direction  (SQ/Sei  and  in  this  case  it  follows  from  equation  (14) 

that  the  group  velocity  associated  with  the  required  value  of  €2  is  along  («i  ru). 
For  light  damping  this  result  will  be  substantially  unaltered,  although  damping 
will  have  an  important  effect  on  the  value  of  the  exponent  -\{€^n^-\-e2iv^  at  the 
stationary  point.  This  effect  can  be  investigated  by  noting  initially  that 

d{e^n^+&^n^)IBr}~{deJbri  B&Jbr]).{n^  n^.  (15) 

Now  it  follows  from  equation  (11)  that  for  light  damping  {ri<l) 

(dQ/de,  dQlde,).{de,ldr]  de^/dv)  =  -io)l2,  (16) 


and  hence  equations  (14)-(16)  can  be  combined  to  yield  the  following  result 
at  the  stationary  point 

d{e^n^+e^n2)/dr)  =  -io)n/2c^.  (17) 


1350 


Here  Cg=^{{d£l!dex?+{d^lbe^'^]  is  the  resultant  group  velocity  and 
n=V[ni+ni-'\  is  the  radial  distance  (in  units)  from  the  excitation  point  to  the 
unit  under  consideration.  It  follows  that  in  the  immediate  vicinity  of  the 
stationary  point  the  exponent  can  be  expanded  in  the  form 


(18) 

where  the  subscript  0  indicates  that  the  term  is  to  be  evaluated  at  the 
stationary  point  under  the  condition  7]=0;  for  ease  of  notation,  this  subscript 
is  omitted  in  the  following  analysis.  The  method  of  steepest  descent  proceeds 
by  substituting  equation  (18)  into  equation  (12)  and  assuming  that:  (i)  the  main 
contribution  to  the  integral  arises  from  values  of  62  in  the  immediate  vicinity 
of  the  stationary  point;  (ii)  the  integrand  is  effectively  constant  in  this  vicinity, 
other  than  through  variation  of  the  term  e2-(e2)o  which  appears  in  equation 
(18);  (iii)  under  conditions  (i)  and  (ii)  the  integration  range  can  be  extended 
to  an  infinite  path  without  significantly  altering  the  result.  The  method  then 
yields  [7] 

w{n,x)  =  ~if  *F7o[20V|aQ/a£j/2xp(A:)p(A:o)|«i(aV5e2)|  . 


(19) 

where/ is  the  normalized  complex  wave  amplitude  which  appears  in  equation 
(9),  and  all  terms  are  to  be  evaluated  at  the  stationary  point. 

The  stationary  point  associated  with  equation  (19)  is  that  point  for 
which  the  group  velocity  is  in  the  {n^  rQ  direction.  Geometrically,  this  is  the 
point  at  which  the  normal  to  the  curve  a)=Q(ei,  £2)  itt  the  plane  lies  in  the 
{n^  Ho)  direction.  Three  such  curves  are  shown  schematically  in  Figure  4, 
together  with  a  specified  (n,  722)  direction.  For  the  frequencies  and  coj  the 
situation  is  straight  forward,  in  the  sense  that  a  unique  stationary  point  exists 
for  any  {n^  n^)  direction.  For  the  frequency  0J2  the  situation  is  more  complex, 
since:  (i)  two  stationary  points  occur  for  the  (n^  rQ  direction  shown,  and  (ii) 
no  stationary  point  exists  if  the  {n^  direction  lies  beyond  the  heading  B 
shown  in  the  figure  (the  dashed  arrow  represents  the  normal  with  maximum 
inclination  to  the  axis).  In  case  (i)  equation  (19)  should  be  summed  over 
the  two  stationary  points,  while  in  case  (ii)  the  method  of  steepest  descent 
predicts  that  w{n,x)  will  be  approximately  zero,  leading  to  a  region  of  very 
low  vibrational  response.  If  the  direction  («i  coincides  with  the  dashed 
arrow,  then  equation  (19)  breaks  down,  since  it  can  be  shown  that 
at  this  point.  The  heading  indicated  by  the  dashed  arrow  represents  a  caustic 
[7],  and  the  theory  given  in  the  present  section  must  be  modified  for  headings 


1351 


(Wi  Wj)  which  are  in  the  immediate  vicinity  of  the  caustic  -  full  details  of  the 
appropriate  modifications  are  given  in  reference  [4].  An  example  which 
illustrates  the  application  of  equation  (19)  is  given  in  section  4. 

3.  RESPONSE  TO  AN  IMPULSIVE  POINT  LOAD 

If  the  system  is  subjected  to  an  impulsive  (i.e.  a  delta  function  applied  at 
r=0),  rather  than  harmonic,  point  load,  then  equation  (10)  becomes  [8] 


w{n,x,t)={N^NJ2Tp)  f  [  g* 

II  (20) 


where  co  =  ^](€i,€2)-  The  method  of  stationary  phase  can  be  applied  to  this 
expression  to  yield  [8] 

w{n,x,t)-{\l2yr)[p{x)p{x)\J\r‘^^^^ 

exp(-/Gjtti  -k^n^nQt+ib) } , 


(21) 

where  all  terms  are  evaluated  at  the  stationary  point,  and  J  and  5  are  defined 
as 


/=(a"n/a6?)(a"Q/ae^-(3"0/ae,a6,)",  S=(Tr/4)sgn(a%/fle?){l+sgn(/)}. 

(22,23) 

In  this  case  the  stationary  point  is  given  by  the  solution  to  the  equations 

=(afi  =(30  /3e,>.  (24,25) 

In  practice  equations  (24)  and  (25)  may  yield  multiple  solutions  (stationary 
points),  in  which  case  equation  (21)  should  be  summed  over  all  such  points. 
Furthermore,  stationary  points  having  7=0  indicate  the  occurrence  of  a 
caustic,  and  equation  (21)  must  be  modified  in  the  immediate  vicinity  of  such 
points  as  detailed  in  reference  [8].  An  example  of  the  application  of  equation 
(21)  is  given  in  the  following  section. 


1352 


4.  EXAMPLE  APPLICATION 


4. 1  The  System  Considered 

The  foregoing  analysis  is  applied  in  this  section  to  a  two-dimensional  periodic 
structure  which  consists  of  a  rectangular  grid  of  lumped  masses  m  which  are 
coupled  through  horizontal  and  vertical  shear  springs  of  stiffness  and  h 
respectively.  Each  mass  has  a  single  degree  of  freedom  consisting  of  the  out- 
of-plane  displacement  w,  and  a  linear  spring  of  stiffness  k  is  attached  between 
each  mass  and  a  fixed  base.  It  is  readily  shown  that  the  system  has  a  single 
phase  constant  surface  of  the  form 

Q  -(ep€,)=iLii(l-cos€i)+/x2(l-cos€2)+a;^, 


where  ix^^lkjm,  and  o)^-==klm.  The  function  U  can  be  used  in 

conjunction  with  the  analysis  of  the  previous  sections  to  yield  the  response  of 
the  system  to  harmonic  and  impulsive  point  loading;  in  this  regard  it  can  be 
noted  that  for  the  present  case  p{x)=m,  V—l,  and /(:»:)  =  1. 

4.2  Response  to  Harmonic  Loading 

The  surface  is  shown  as  a  contour  plot  in  Figure  5  for  the  case 

m=1.0,  oj„==0,  Ati  =  1.0,  Results  for  the  forced  harmonic  response 

of  this  system  at  the  two  frequencies  w  =  1.003  and  cx>  =  1.181  are  shown  in 
Figures  6  and  7.  In  each  case  the  response  of  a  40  x40  array  of  point  masses 
is  shown;  a  unit  harmonic  point  load  is  applied  to  mass  (21,21)  and  the  loss 
factor  is  taken  to  be  =0.05.  Two  sets  of  contours  are  shown  in  each  Figure: 
the  smooth  contours  have  been  calculated  by  using  equation  (19)  while  the 
more  irregular  contours  have  been  obtained  by  a  direct  solution  of  the 
equations  of  motion  of  the  finite  1600  degree-of-freedom  system.  By 
considering  the  results  shown  in  Figure  6,  it  can  be  concluded  that:  (i)  for  the 
present  level  of  damping  the  finite  system  effectively  behaves  like  an  infinite 
system,  and  (ii)  the  analytical  result  yielded  by  equation  (19)  provides  a  very 
good  quantitative  estimate  of  the  far  field  response.  It  can  be  noted  from 
Figure  5  that  no  caustic  occurs  for  a)  =  1.003,  in  the  sense  that  equation  (14) 
yields  only  one  stationary  point  which  contributes  to  equation  (19).  In 
contrast,  a  caustic  does  occur  for  the  case  a)  =  1.181,  and  this  leads  to  the  very 
irregular  spatial  distribution  of  response  shown  in  Figure  7.  Two  stationary 
points  contribute  to  equation  (19),  and  constructive  and  destructive 
interference  between  these  contributions  is  responsible  for  the  rapid 
fluctuations  in  the  response  amplitude.  It  is  clear  that  the  response  exhibits 
a  "dead  zone"  for  points  which  lie  beyond  the  caustic  heading  (in  this  case 
30.25°  to  the  «i-axis),  as  predicted  by  the  analysis  presented  in  section  2. 

4.3  Response  to  Impulsive  Loading 

The  impulse  response  of  a  system  having  m=1.0,  /xi  =  1.0,  ^2=0. 51,  and 


1353 


co„'“0.25  has  been  computed.  The  impulse  was  taken  to  act  at  the  location 
/Zi=«2=0  and  the  time  history  of  the  motion  of  each  mass  in  the  region  - 
iO<(72i,n2)  ^  10  was  found  by  using  equations  (21)-(23).  For  each  mass  the 
maximum  response  \w\  was  recorded,  and  the  results  obtained  are  shown  as 
a  contour  plot  in  Figure  8.  In  accordance  with  Fourier’s  Theorem,  the 
impulse  response  of  the  system  contains  contributions  from  all  frequencies, 
and  therefore  the  spatial  distribution  of  |  w|  can  be  expected  to  lie  somewhere 
between  the  two  extreme  forms  of  harmonic  response  exhibited  in  Figures  6 
and  7.  This  is  in  fact  the  case,  and  the  response  shown  in  Figure  8  retains  a 
distinctive  spatial  pattern.  As  discussed  in  reference  [8],  the  results  shown  in 
Figure  8  are  in  good  agreement  with  direct  simulation  of  the  impulse  response 
of  the  system. 


5.  CONCLUSIONS 

This  paper  has  considered  the  response  of  a  two-dimensional  periodic  structure 
to  both  harmonic  and  impulsive  point  loading.  With  regard  to  harmonic 
loading,  it  has  been  shown  that  the  spatial  pattern  of  the  response  is  strongly 
dependent  on  the  occurrence  of  a  caustic:  if  no  caustic  occurs  then  the 
response  has  a  fairly  smooth  spatial  distribution,  whereas  the  presence  of  a 
caustic  leads  to  an  irregular  spatial  distribution  and  a  "dead  zone"  of  very  low 
response.  This  type  of  feature  is  also  exhibited,  although  to  a  lesser  degree, 
in  the  spatial  distribution  of  the  response  to  an  impulsive  point  load.  This 
behaviour  could  possibly  be  exploited  to  reduce  vibration  transmission  along 
a  specified  path,  although  the  practicality  of  this  approach  for  a  complex 
system  has  yet  to  be  investigated.  The  present  analytical  approach  can  be 
applied  to  all  types  of  two-dimensional  periodic  structure  -  the  information 
required  consists  of  the  phase  constant  surface(s)  0(61,62)  and  the  associated 
wave  form(s)  f{x),  both  of  which  are  yielded  by  standard  techniques  for  the 
analysis  of  free  wave  motion  in  periodic  structures  [1,2]. 

REFERENCES 

1.  S.S.  MESTER  and  H.  BENAROYA  1995  Shock  and  Vibration  2,  69- 
95.  Periodic  and  near-periodic  structures. 

2.  D.J.  MEAD  1996  Journal  of  Sound  and  Vibration  190,  495-524. 
Wave  propagation  in  continuous  periodic  structures:  research 
contributions  from  Southampton  1964-1995. 

3.  L.  MEIROVITCH  1986  Elements  of  Vibration  Analysis,  Second 
Edition.  New  York:  McGraw-Hill  Book  Company. 

4.  R.S.  LANGLEY  1996  Journal  of  Sound  and  Vibration  (to  appear). 
The  response  of  two-dimensional  periodic  structures  to  point  harmonic 
forcing. 


1354 


1 

■ 

■ 

■ 

1 

■ 

■ 

■ 

0 

m 

■ 

1 

■ 

Xj.  ■j' 

Oi 

< - 

^1 

1  ^ 

Figure  I ,  Schematic  of  a  two-dimensional  periodic  structure.  The  arrow  indicates'  the  reference  unit 
(with  n=0)  while  the  circle  represents  a  general  point  (re,x).  The  structure  may  have  a  third  spatial 
coordinate  X3,  which  for  convenience  is  not  shown  in  the  figure.  The  point  load  considered  in  section 
3  is  applied  at  the  location  of  the  arrow. 


Figure  2.  Phase  constant  surfaces  for  a  plate  which  rests  on  a  square  grillage  of  simple  supports. 
Q  is  a  non-dimensional  frequency  which  is  def  as  Q=a)LV(m/D),  where  m  and  D  are  respectively 
the  mass  per  unit  area  and  the  flexural  rigidii^  the  plate,  and  L  is  the  support  spacing. 


1355 


Figure  5.  Contour  plot  of  the  phase  constant  surface  for  the  case  ^=0.57.  The  contours 

are  separated  by  an  increment  Aco =0. 1477.  The  two  contours  considered  in  section  4.2  are  indicated 
as  follows:  (a)  a)  =  1.033;  (b)  a)  =  1.181. 


Figure  6.  Response  |H'(n,x)|2  of  the  40x4v  .uass/spring  system  to  a  unit  harmonic  force  of 
frequency  w- 1.033  applied  at  the  location  i=y=21.  The  contours  correspond  to  the  response  levels 
lK«,x)!'=0.01,  0.02,  andO.05. 


1357 


STICK-SLIP  MOTION  OF  AN  ELASTIC  SLIDER  SYSTEM  ON 
A  VIBRATING  DISC 


HOuyang  J  E  Mottershead  M  P  CartmeU '  MIFiiswell^ 
Department  of  Mechanical  Engineering,  University  of  Liverpool 
^  Department  of  Mechanical  Engineering,  University  of  Edinburgh 
^  Department  of  Mechanical  Engineering,  University  of  Wales  Swansea 

ABSTRACT 


The  in-plane  vibration  of  a  slider-mass  which  is  driven  around  the  surface  of  a 
flexible  disc,  and  the  transverse  vibration  of  the  disc,  are  investigated.  The  disc 
is  taken  to  be  an  elastic  annular  plate  and  the  slider  has  flexibility  in  the 
circumferential  (in-plane)  and  transverse  directions.  The  static  fiiction 
coefi&cient  is  assumed  to  be  higher  than  the  kinetic  friction.  As  a  result  of  the 
fiiction  force  acting  between  the  disc  and  the  slider  system,  the  slider  will 
oscillate  in  the  stick-slip  mode  in  the  plane  of  the  disc.  The  transverse  vibration 
induced  by  the  slider  will  change  the  normal  force  of  the  slider  system  acting 
on  the  disc,  which  in  turn  will  change  the  in-plane  oscillation  of  the  slider.  For 
different  values  of  system  parameters,  the  coupled  in-plane  oscillation  of  the 
slider  and  transverse  vibration  of  the  disc  will  exhibit  quasi-periodic  as  well  as 
chaotic  behaviour.  Rich  patterns  of  chaotic  vibration  of  the  slider  system  are 
presented  in  graphs  to  illustrate  the  special  behaviour  of  this  non-smooth 
nonlinear  dynamical  system  The  motivation  of  this  work  is  to  analyse  and 
understand  the  instability  and/or  squeal  of  physical  systems  such  as  car  brake 
discs  where  there  are  vibrations  induced  by  non-smooth  dry-fiiction  forces. 


NOMENCLATURE 


a ,  b  mner  and  outer  radii  of  the  annular  disc 
c  damping  coefficient  of  the  slider  in  in-plane  direction 
h  thickness  of  the  disc 


i  =  V=T 


1359 


k ,  k  transverse  and  in-plane  stiffiiess  of  the  slider  system 


m 

r 


t 


^stiek 


mass  of  the  slider 

radial  co-ordinate  in  cylindrical  co-ordinate  system 
radial  position  of  the  slider 

modal  co-ordinate  for  k  nodal  circles  and  /  nodal  diameters 
time 

the  time  of  the  onset  of  sticking 
u ,  Mp  transverse  displacement  of  the  slider  mass  and  its  initial  value 
w ,  Wq  transverse  displacement  of  the  disc  and  its  initial  value 
flexural  rigidity  of  the  disc 
Kelvin-type  damping  coefficient 
Young’s  modulus 

initial  normal  load  on  the  disc  jfrom  the  slider  system 
total  normal  force  on  the  disc  from  the  slider  system 
combination  of  Bessel  functions  representing  mode  shape  in  radial 
direction 

circumferential  co-ordinate  of  cylindrical  co-ordinate  system 

kinetic  and  static  dry  fiction  coefficient  between  the  shder  and  the 


D 

D' 

E 

N 

P 

R. 


0 


disc 


V 

P 

(P 

^  stick 
¥ 
¥ki 


CO. 


Poisson’s  ratio 

damping  ratio  of  the  disc 

specific  density  of  the  disc 

absolute  circumferential  position  of  the  slider 

absolute  circumferential  position  of  the  slider  when  it  sticks  to  the  disc 
circumferential  position  of  the  slider  relative  to  the  drive  point 
mode  function  for  the  transverse  vibration  of  the  disc  corresponding  to 

^kl 

natural  (circular)  frequency  correq)onding  to  <5^^ 


1360 


Q  constant  rotating  speed  of  the  drive  point  around  the  disc  in  radians  per 
second 


INTRODUCTION 


There  exists  a  whole  class  of  mechanical  systems  which  involve  discs  rotating 
relative  to  stationery  parts,  such  as  car  brake  discs,  clutches,  saws,  computer 
discs  and  so  on.  In  these  systems,  dry-friction  induced  vibration  plays  a  crucial 
role  in  system  performance.  If  the  vibration  becomes  excessive,  the  system 
might  fail,  or  cease  to  perform  properly,  or  make  offensive  noises.  In  this 
paper,  we  investigate  the  vibration  of  an  m-plane  slider  system,  with  a 
transverse  mass- spring- damper,  attached  through  an  in-plane  spring  to  a  drive 
point  which  rotates  at  constant  speed  around  an  elastic  disc,  and  the  vibrations 
of  the  disc.  Dry  friction  acts  between  the  sHder  system  and  the  disc. 

Dry-friction  induced  vibration  has  been  studied  extensively  [1-4].  For  car  brake 
vibration  and  squeal,  see  the  review  papers  [5,6].  The  stick-slip  phenomenon  of 
dry-friction  induced  vibration  is  studied  in  the  context  of  chaotic  vibration  [7- 
10].  Popp  and  Stelter  [7]  studied  such  motion  of  one  and  two  degrees  of 
freedom  system  and  foimd  chaos  and  bifiircation.  They  also  conducted 
experiments  on  a  beam  and  a  circular  plate  (infinite  number  of  degrees  of 
freedom).  These  theoretical  works  are  about  systems  of  less  than  three  degrees 
of  freedom,  and  the  carrier  which  activates  the  friction  is  assumed  to  be  rigid. 
In  this  paper,  we  consider  an  elastic  disc  so  that  the  transverse  vibrations  of  the 
disc  are  important.  As  a  result  of  including  the  transverse  vibrations  of  the  disc, 
rich  patterns  of  chaos,  which  have  not  been  reported  previously  are  found.  If 
there  is  only  shding  present  at  constant  speed,  the  problem  is  reduced  to  a 
linear  parametric  analysis  which  was  carried  out  for  a  pin-on- disc  system  in 
[11]  and  for  a  pad-on-disc  system  in  [12,13]. 

m-PLANE  OSCILLATION  OF  THE  SLIDER  SYSTEM 


As  the  drive  point,  which  is  connected  to  the  shder-mass  through  an  in-plane, 
elastic  spring,  is  rotated  at  constant  angular  speed  around  the  disc,  the  driven 
slider  will  undergo  stick-slip  oscillations.  The  whole  system  of  the  shder  and 
the  disc  is  shown  in  Figure  1. 

The  equation  of  the  in-plane  motion  of  the  slider  system  relative  to  the  rotating 
drive  point,  in  the  sliding  phase,  is. 


1361 


(1) 


while  in  sticking,  the  equation  of  the  motion  becomes. 

The  relationship  between  the  relative  motion  of  the  slider  system  to  the  drive 
point  and  its  absolute  motion  (relative  to  the  stationary  disc)  is 

(p  =  Qt  +  y/,  (3) 

We  consider  the  foEowing  initial  conditions  which  are  intended  to  simulate 
what  happens  in  a  disc  brake.  The  slider  system  is  at  rest  and  there  is  no  normal 
loading  on  the  disc  j&om  the  slider.  Then  a  constant  normal  load  is  applied 
which  causes  transverse  vibrations  in  the  disc.  At  the  same  time,  the  drive  point 
is  given  a  constant  angular  velocity.  Other  initial  conditions  are  possible,  so 
that  there  is  no  loss  of  generality. 

First,  sliding  from  the  initial  sticking  phase  occurs  when, 

The  slider  will  stick  to  the  disc  agaiu  when, 

ju^P  (during  sliding),  (5) 

or  it  will  begin  to  sHde  again  iJ^ 

ij/  =  ,  \k^rQtp-\  >  (dxiring  sticking).  (6) 

Consequently,  the  slider  system  will  stick  and  slide  consecutively  on  the  disc 
surface. 


TRANSVERSE  VIBRATION  OF  THE  ANNULAR  DISC 

The  equation  of  motion  of  the  disc  under  the  slider  system  is, 

ph^  +  D'V^'w  +  DW  =  --5(r-r„)5i$-<p)P.  (7) 

a/"  r 

The  total  force  P  is  the  summation  of  initial  normal  load  N  and  the  resultant 
of  the  transverse  motion  w  of  the  slider.  Its  expression  is, 

P  -  N +  mu  +  cu-h kiu-u^) .  (8) 

Since  it  is  assumed  that  the  slider  system  is  always  in  perfect  contact  with  the 
disc,  then, 


1362 


(9) 


u{t)  =  w{r„(p{t)A- 

Substitution  of  ecpiations  (8)  and  (9)  into  (7)  leads  to. 


ph 


^  —  +  /:)VV  =  --5(r  -  r,  )6(^  -  (p)[N  + 


dt 


dt 


^..dw  d^w  d^w  .dw  dw 


(10) 


k(w-wj\. 

Note  that  equation  (10)  is  valid  whether  the  slider  system  is  sticking  or  sliding. 
When  the  slider  sticks  to  the  disc,  equation  (10)  reduces  to. 


+  D-V‘  ^ +£)W  =  --5(r  - )5(5 -<p)x 

St  St  r  (11) 

d'^W  ^  M 

[A^+m— +  c— +  A:(w-w„)]. 


COUPLED  VIBRATIONS  OF  THE  SLIDER  AND  THE  DISC 


Assume  that  the  transverse  motion  of  the  disc  can  be  represented  by, 

M>{r,e,t)=ttii/,{r,0)q,it),  (12) 

Jt=0  /=-<io 


and, 

where  {r)  is  a  combination  of  Bessel  functions  satisfying  the  boundary 
conditions  in  radial  direction  at  the  inner  radius  and  outer  radius  of  the  disc. 
The  modal  functions  satisfy  the  ortho-normality  conditions  ofj 


(14) 


1363 


Equations  (10)  and  (11)  can  be  simplified  by  being  written  in  terms  of  the 
modal  co-ordinates  from  equation  (12). 

During  sticking,  the  motion  of  the  whole  system  of  the  slider  and  the  disc  can 
be  represented  by, 

N 

+  2^0) - 
— Z^„('-o)^«('-o)exp[i(5-0(»]x  (15) 

ph.U  ''=0  s=-«o 

{mq„+cq^+k{q^-q„{^)\). 


The  sticking  phase  can  be  maintained  i^ 


rA\¥\</^XJ^+- 


^Z  Z  ^«(?')exp(i/?»)x 


{mq„+cq„+  k[q„  -  (0)] }  ] . 

While  in  sliding,  the  motion  of  the  whole  system  can  be  represented  by. 


%+2^a>^,qu+a>lq„  = 


^^«('i)exp(-i/<»)- 


-rrri  Z^„('-o)^«(n,)exp[i(5-0«!’]x 

pnO  r=<i  5^=-oo 

H?,.  +i2sw„  +isw„)  + 

k{q„-qS'^)]}, 


r„{m{j/  +  kw)  =  -/i,Sign((Z))[A''  +- 


^Z  Z  KMx 


exp(LS(Z>)M?„  +i2s^„  +(ls^-5>^)9„]  + 

c{q„  +>s(?g„) +*[!?„  -?„(o)]}]. 

The  sliding  phase  can  be  maintained  if, 


1364 


\K^M< mXN +-J^t  t  K{r,)es.p{is(p)x 
■yjphb  ^=0^^ 

{iTiq„+i2sj>q„+(is^-s^j)^)qJ+  (20) 

c(?„  +isj^J  +  k[q„  -9„(0)]}], 

when, 

\j/  =  ~0  or  ^  =  0.  (21) 

COMPUTING  PROCESS 


As  the  shder  system  sticks  and  sHdes  consecutively,  the  governing  equations  of 
the  coupled  motions  of  the  whole  system  switch  repeatedly  from  equations 
(15),  (16)  and  (17)  to  equation  (18),  (19)  and  (20).  TTie  system  is  not  smooth. 
Since  the  condition  which  controls  the  phases  of  the  slider  system  itself 
depends  on  the  motions,  it  is  also  a  nonlinear  system,  whether  is  a  constant 
or  a  function  of  relative  speed  ^ .  In  order  to  get  modal  co-ordinates,  we  have 
to  truncate  the  mfinite  series  in  equation  (12)  to  jfinite  terms.  Then  numerical 
integration  is  used  to  solve  equations  (15),  (18)  and  (19).  Here  a  fourth  order 
Rimge-Kutta  method  is  used  for  second  order  simultaneous  ordinary 
differential  equations. 

Since  equation  (18)  has  time-dependent  coefficients,  time  step  length  has  to  be 
very  small.  Constant  time  step  lengths  are  chosen  when  the  m-plane  slider 
motion  is  well  within  the  sticking  phase  or  the  shding  phase  in  the  numerical 
integration.  As  it  is  imperative  that  the  time  step  should  be  chosen  such  that  at 
the  end  of  some  time  intervals  the  shder  happens  to  be  on  the  sticking- shding 
interfaces,  we  use  a  prediction  criterion  to  choose  next  time  step  length  when 
approaching  these  interfaces.  Therefore,  at  the  sticking-shding  interfaces,  the 
time  step  length  is  variable  (actuahy  smaher  than  it  is  while  weh  within  sticking 
or  shding).  Nevertheless,  tbe  interfaces  equations  (  equation  (17)  or  equations 
(20-21) )  are  only  approximately  satisfied  [10]. 

When  transverse  motion  of  the  shder  system  becomes  so  violent  that  the  total 
normal  force  P  in  it  becomes  negative  or  becomes  several  times  larger  than 
the  initial  normal  load  7/ ,  we  describe  the  system  as  being  unstable.  Then  the 
motion  begins  to  diverge.  But  this  instabihty  should  be  distinguished  from  a 
chaotic  motion  which  is  bounded  but  never  converge  to  a  point. 


1365 


NUMERICAL  EXAMPLES 


The  following  data  are  used  in  the  computation  of  numerical  examples: 
a  =  0.065m,  b  =  012m,  OTm, /z  =  0.001m;  =  120GPa,  v=  0.35, 

Z)*=  0.00004;  yW,  =0.4,  //^=0.24,  A:  =  lOOON/m,  =  lOON/m, 

m  =  0.1kg,  p  =  7000kg / m^  The  disc  is  clamped  at  inner  radius  and  free  at 

outer  radius.  Note  that  in  these  numerical  examples,  the  disc  thickness  is 
dehherately  taken  to  be  very  small  in  order  to  reduce  the  amount  of  computing 
work.  However,  this  will  not  affect  the  qualitative  features  of  the  results  or 
conclusion  drawn  from  the  results  thus  obtained.  The  first  five  natural 
(circular)  frequencies  are  451.29,  462.73,  426.73,  508.23,  508.23.  We  will 
concentrate  on  the  vibration  solutions  at  different  levels  of  initial  normal  load. 
But  occasionally  solutions  at  different  rotating  speed  or  different  damping 
ratios  are  investigated.  Unless  specified  expressly,  the  Poincare  sections  are  for 
the  in-plane  vibration  of  the  slider  system 

First  of  all,  we  study  the  effect  of  the  normal  loadA^.  Take  f2=l0  and 
^ —  When  N  is  very  small^  the  Poincare  section  is  a  perfect  ellipse 

which  indicate  the  in-plane  vibration  of  the  slider  system  is  quasi-periodic,  as 
the  transverse  vibration  of  the  disc  is  too  small  to  affect  total  normal  force  P . 
A  typical  plot  of  such  motion  is  shown  in  Figure  2  for  A^=0.5kPa.  As  N 
increases,  the  sticking  period  gets  longer,  the  bottom  part  of  the  elhpse  evolves 
into  a  straight  line,  indicating  phase  points  within  the  sticking  phase.  One  of 
such  plots  is  given  in  Figure  3  for  N  =3kPa.  A  further  increase  of  N  not  only 
lengthens  the  straight  line  part  of  the  Poincare  section,  but  also  creates  an 
increasingly  ragged  outline  in  the  arch  part  of  the  plot.  The  curve  is  no  longer 
smooth  and  it  seems  that  the  in-plane  motion  begins  to  enter  a  chaotic  state 
from  the  quasi-periodic  state.  Figure  4  presents  the  Poincare  section  plot  for 
N  =7.5kPa.  There  is  a  transition  period  from  quasi-periodic  motion  to  chaotic 
motion,  extending  from  N  =6kPa  up  to  N  =9kPa.  Chaos  becomes  detectable 
at  iV=10kPa,  whose  Poincare  section  is  shown  in  Figure  5.  Then  chaotic 
vibration  follows.  When  N  ~15kPa,  the  arch  part  of  the  Poincare  section 
becomes  so  fuzzy  and  thick  that  it  should  no  longer  be  considered  as  a  curve, 
but  rather  a  narrow  (fractal)  area.  A  hlow-up’  view  of  the  arch  part  reveals 
that  phase  points  are  distributed  across  the  arch.  Both  plots  are  ^own  in 
Figure  6.  Between  iV'=17.ikPa  and  18.325kPa,  the  vibration  of  the  slider 
enters  a  new  stage,  with  Poincare  sections  looking  like  star  clusters  as 
illustrated  in  Figures  7  and  8.  This  kind  of  motions  are  rather  extraordinary  and 


1366 


have  not  been  reported  in  other  works  on  stick-shp  motions  with  a  rigid 
carrier.  Afterwards,  the  ‘arch-door’  hke  Poincare  sections  come  back  (see 
Figure  9).  The  difference  from  previous  Poincare  sections  of  lower  N  is  that 
the  new  Poincare  sections  look  like  overlapping  of  earlier  Poincare  sections, 
which  indicates  a  clear  layered  structure,  as  diown  in  Figure  10  and  more 
obviously  in  the  left  hand  side  of  Figure  11.  At  this  stage,  the  vibration  is  very 
chaotic.  To  give  the  reader  a  better  picture,  the  Poincare  section  of  a  fixed 
point  on  the  disc  at  ( =  0.1m  and  =  0 ),  is  also  shown  in  the  right  hand  side 
of  Figure  11.  The  Poincare  sections  of  the  slider-mass  and  a  point  on  the  disc 
are  also  given  in  Figures  12-15.  In  Figure  12  for  A^=30.5kPa,  the  vibration 
goes  unstable.  Here  again,  the  Poincare  sections  have  not  been  reported 
elsewhere. 

If  disc  damping  is  increased,  vibration  will  become  more  regular,  as  shown  in 
Figure  13.  Comparing  Figures  11  with  13,  we  see  that  increase  of  disc  damping 
makes  the  vibration  more  concentrated  though  not  always  smaller.  Unstable 
vibration  can  be  stabilised  with  more  disc  damping,  as  seen  from  Figure  14. 

If  there  is  no  damping  at  aU,  the  resulting  vibration  due  to  dry  fiiction  will  be 
unstable,  even  at  very  small  normal  load  N .  In  Figure  15,  the  motion  of  the 
slider  tends  to  run  away  in  the  tangential  direction  from  the  normal  ellipse 
attractor,  while  the  motion  of  the  disc  goes  unbounded. 

Increasing  the  speed  of  the  drive  point  seems  to  make  vibration  more  chaotic 
and  more  unstable,  as  shown  in  Figures  16-18.  At  this  stage,  however,  we  are 
unable  to  make  a  definite  conclusion  on  rotating  speed  as  there  might  be 
intervals  of  regular  motions  and  intervals  of  chaotic  motion  for  .  More 
numerical  examples  must  be  computed  to  draw  a  positive  conclusion  on  this 
parameter. 

The  correlation  dimension  is  not  a  good  measure  of  the  vibration  for  the 
current  problem  because  its  values  fluctuate  in  some  numerical  examples.  This 
failure  was  perhaps  first  discovered  in  [7].  The  reason  can  be  either  that  the 
system  is  non-smooth,  or  that  the  system  has  multiple  degrees  of  freedom,  or 
both.  Therefore,  the  correlation  dimension  or  any  other  fractal  dimensions  is 
not  presented  in  this  paper. 


CONCLUSIONS 


In  this  paper,  we  studied  the  in-plane  stick-slip  vibration  of  a  slider  system  with 
a  transverse  mass-spring-damper  driven  around  an  elastic  disc  through  a  spring 
from  a  constant  speed  drive  point,  and  transverse  vibrations  of  the  disc.  The 


1367 


whole  system  had  been  reduced  to  six  degrees  of  freedom  after  simplification. 
From  numerical  examples  computed  so  far,  we  can  conclude  that: 

1.  Both  vibrations  are  very  complex  as  this  is  a  multi-degree  of  freedom,  non¬ 
smooth  system  Rich  patterns  of  chaotic  vibration  are  found.  Some  have  not 
been  reported  elsewhere. 

2.  For  the  normal  pressure  parameter,  smaller  values  allow  quasi-peiiodic 
solutions.  Greater  pressures  result  in  chaotic  motions.  At  certain  large 
pressures,  the  vibrations  become  unstable. 

3.  Disc  dartping  makes  vibration  more  concentrated  to  smaller  areas  and  when 
sufficiently  large  it  can  stabilise  otherwise  unstable  vibration. 

4.  An  increase  in  the  rotating  speed  can  make  the  vibration  more  chaotic  or 
more  unstable. 

5.  Correlation  dimension  is  not  a  good  measure  of  the  vibration  of  this  multi¬ 
degree  of  freedom,  non-smooth  dynamical  system 

6.  Much  more  investigation  needs  to  be  earned  out  in  understanding  and 
characterising  the  vibration  of  multi- degree  of  freedom,  non- smooth  dynamical 
systems. 


ACKNOWLEDGEMENT 

This  research  is  supported  by  the  Engineering  and  Physical  Sciences  Research 
Council  (grant  niunber  J35177)  and  BBA  Friction  Ltd. 

REFERENCES 


1.  Nakai,  M.,  Chiba,  Y.  and  Yokoi,  M.,  Railway  wheel  squeal.  Bulletin  of 
JSME,  1984,  27,  301-8 

2.  Lin,  Y-Q  and  Wang  Y-H,  Stick-sHp  vibration  of  drill  strings.  XEng.Ind., 
TransASME,  1991, 113,  38-43 

3.  Ferri,  A.  A.  and  Bindemann,  A.C.,  Damping  and  vibration  of  beams  with 
various  types  of  fiictional  support  conditions.  J.  VibAcoust,  TransASME^ 
1992,  114,  289-96 

4.  Lee,  A.C.,  Study  of  disc  brake  noise  using  multi-body  mechanism  with 
friction  interface.  In  Friction-Induced  Vibration,  Chatter,  Squeal,  and 
Chaos,  Ed.  Ibrahim,  KA.  and  Soom,  A.,  DE-Vol.  49,  ASME  1992,  pp.99- 
105 


1368 


5.  Ibrahim,  R.A.,  Friction-induced  vibration,  chatter,  squeal,  and  chaos.  In 
Friction-Induced  Vibration,  Chatter,  Squeal,  and  Chaos,  Ed.  Ibrahim, 
KA.  and  Soom,  A,  DE-VoL  49,  ASME  1992,  pp.  107-38 

6.  Yang,  S.  and  Gibson,  R.F.,  Brake  vibration  and  noise:  reviews  ,comments, 
and  proposed  considerations.  Proceedings  of  the  14th  Modal  Analysis 
Conference,  The  Society  of  Experimental  Mechanics,  Inc.,  1996,  pp.l342- 
9 

7.  Popp,  K  and  Stelter,  P.,  Stick-slip  vibration  and  chaos.  Phil.  Trans.  R  Soc. 
Lond.  A(1990),  332,  89-106 

8.  Pfeiffer,  F.  and  Majek,  M.,  Stick-slip  motion  of  turbine  blade  dampers.  Phil. 
Trans.  R  Soc.  Lond.  A(1992),  338,  503-18 

9.  Wojewoda,  J.,  Kapitaniak,  T.,  Barron,  R.  and  Brindley,  J.,  Complex 
behaviour  of  a  quasiperiodically  forced  experimental  system  with  dry 
friction.  Chaos,  Solitons  and  Fractals,  1993,  3,  35-46 

10.  Wiercigroch,  M.,  A  note  on  the  switch  function  for  the  stick-slip 
phenomenon.  J.SoundVib.,  1994, 175,  700-4 

11.  Chan,  S.N.,  Mottershead,  J.E.  and  Cartmell,  M.P.,  Parametric  resonances 
at  subcritical  speeds  in  discs  with  rotating  frictional  loads.  Proc.  Instn. 
Mech.  Engrs,  1994,  208,  417-25 

12.  Mottershead,  J.E.,  Ouyang,  R,  Cartmell,  M.P.  and  Friswell,  M.I., 
Parametric  Resonances  in  an  annular  disc,  with  a  rotating  system  of 
distributed  mass  and  elasticity;  and  the  effects  of  friction  and  dan:q)ing. 
Proc.  Royal  Soc.  Lond.  A.,  1997,  453,  1-19 

13.  Ouyang,  H.,  Mottershead,  J.E.,  Friswell,  M.l.  and  Cartmell,  M.P.,  On  the 
prediction  of  squeal  in  automotive  brakes.  Proceedings  of  the  14th  Modal 
Analysis  Conference,  The  Society  of  Experimental  Mechanics,  Inc.,  1996, 
pp.  1009-16 


Figure  1.  Slider  system  and  disc  in  cylindrical  co-ordinate  system 


1369 


Figure  9.  iV=21kPa 


Figure  10.  iV=24kPa 


1374 


A  Finite  Element  Time  Domain 
Multi-Mode  Method  For  Large  Amplitude 
Free  Vibration  of  Composite  Plates 

Raymond  Y.  Y.  Lee,  Yucheng  Shi  and  Chuh  Mei 
Department  of  Aerospace  Engineering 
Old  Dominion  University,  Norfolk,  VA  23529-0247 


Abstract 

This  paper  presents  a  time-domain  modal  formulation  using  the  finite  element  method  for 
large-amplitude  firee  vibrations  of  generally  laminated  thin  composite  rectangular  plates.  Accurate 
fi'equency  ratios  for  fundamental  as  well  as  higher  modes  of  composite  plates  at  various  maximum 
deflections  can  be  determined.  The  selection  of  the  proper  initial  conditions  for  periodic  plate 
motions  is  presented.  Isotropic  beam  and  plate  can  be  treated  as  special  cases  of  the  composite 
plate.  Percentage  of  participation  from  each  linear  mode  to  the  total  plate  deflection  can  be 
obtained,  and  thus  an  accurate  frequency  ratio  using  a  minimum  number  of  linear  modes  can  be 
assured.  Another  advantage  of  the  present  finite  element  method  is  that  the  procedure  for  obtaining 
the  modal  equations  of  the  general  DujB5ng-type  is  simple  when  compared  with  the  classical 
continuum  Galerkin’s  approach.  Accurate  frequency  ratios  for  isotropic  beams  and  plates,  and 
composite  plates  at  various  amplitudes  are  presented. 


Introduction 

Large  amplitude  vibrations  of  beams  and  plates  have  interested  many 
investigators  [1]  ever  since  the  first  approximate  solutions  for  simply  supported 
beams  by  Woinowsky-Kiieger  [2]  and  for  rectangular  plates  by  Chu  and  Herrmann 
[3]  were  presented.  Singh  et  al.  [4]  gave  an  excellent  review  of  various  formulation 
and  assumptions  ,  including  the  finite  element  method  for  large  amplitude  firee 
vibration  of  beams.  Srirangaraja  [5]  recently  presented  two  alternative  solutions, 
based  on  the  method  of  multiple  scales  (MMS)  and  the  ultraspherical  polynomial 
approximation  (UP A)  method,  for  the  large  amplitude  firee  vibration  of  a  simply 
supported  beam.  The  fi'equency  ratios  for  the  fundamental  mode,  ca/C0L>  at  the  ratio 
of  maximum  beam  deflection  to  radius  of  gyration  of  5.0  (Wmax/r  =5.0)  are  3.3438 
and  3.0914,  using  the  MMS  and  the  UPA  method,  respectively.  Eleven  firequency 
ratios  including  nine  firom  reference  [4]  were  also  given  (see  Table  1  of  reference 
[5]).  It  is  rather  surprising  that  the  firequency  ratio  for  the  fundamental  mode  at 
Wn«x/r  =5.0  for  a  simply  supported  beam  varied  in  a  such  wide  range:  fi-om  the 
lowest  of  2.0310  to  the  highest  of  3.3438,  and  with  the  elliptic  function  solution  by 
Woinowsky-Kiieger  [2]  giving  2.3501.  Similar  wide  spread  exists  for  the  vibration 
of  plates.  Rao  et  al  [6]  presented  a  finite  element  method  for  the  large  amplitude 
firee  flexural  vibration  of  unstiffened  plates.  For  the  simply  supported  square  plate. 


1375 


frequency  ratios  from  six  different  approaches  were  reported  (see  Table  1  of 
reference  [6]).  The  frequency  ratio  at  Wmax/h  =1.0  varied  from  a  low  of  1.2967  to  a 
high  of  1.5314,  with  Chu  and  Herrmann’s  analytical  solution  [3]  at  1.4023. 

This  paper  presents  a  finite  element  time  domain  modal  formulation  for  the  large 
amplitude  free  vibration  of  composite  plates.  The  formulation  is  an  extension  from 
the  isotropic  plates  [7],  and  the  determination  of  initial  conditions  for  periodic 
motions  was  not  employed  in  reference  [7].  The  convergence  of  the  fundamental 
frequency  ratio  is  investigated  for  a  simply  supported  beam  and  a  simply  supported 
square  plate  with  a  varying  number  of  finite  elements  and  a  varying  number  of 
linear  modes.  Accurate  frequency  ratios  for  fundamental  and  higher  modes  at 
various  maximum  deflections,  and  percentages  of  participation  from  various  linear 
modes,  are  obtained  for  beams  and  composite  plates. 


Formulation 

Strain-Displacement  and  Constitutive  Relations 

The  von  Karman  strain-displacement  relations  are  applied.  The  strains  at  any 
point  z  through  the  thickness  are  the  sum  of  membrane  and  change  of  curvature 
strain  components: 


{e}  = 


-w.xx 

V,y 

>+< 

/  2 

>  +  z< 

-W,yy 

U,y+V,x^ 

W,yj 

=  {Em}  +  {sb}  +  z{K} 


(1) 


where  and  {eb}are  the  membrane  strain  components  due  to  in-plane 

displacements  u  and  v  and  the  transverse  deflection  w,  respectively.  The  stress 
resultants,  membrane  force  {N}  and  bending  moment  {M},  are  related  to  the  strain 
components  as  follows: 


■[A]  IB]' 
[B1  [Di; 


(2) 


where  [A]  is  the  elastic  extensional  matrix,  [D]  is  the  flexural  rigidity  matrix,  and 
[B]  is  the  extension  coupling  matrix  of  the  laminated  plate. 

Element  Displacements,  Matrices  and  Equations 

Proceeding  from  this  point,  the  displacements  in  equation  (1)  are  approximated 
over  a  typical  plate  element  ,  e.g.  rectangular  [8]  or  triangular  [9],  using  the 
corresponding  interpolation  functions.  The  in-plane  displacements  and  the  linear 
strains  are  interpolated  from  nodal  values  by 


1376 


where  [Hm]  and  [Bm]  denote  the  displacement  and  strain  interpolation  matrices, 
respectively,  and  {Wm}  is  the  in-plane  nodal  displacement  vector.  The  transverse 
displacement,  slopes  and  curvatures  are  interpolated  from  the  nodal  values  by 

w  =  [Hb]{w^},  |^’^|  =  [G]{wb},  {K}  =  [B|,]{wb}  (4a,b,c) 


where  [HJ  and  [G]  and  [BJ  denote  the  bending  displacement,  slope  and  curvature 
interpolation  matrices,  respectively,  and  {wb  }  denotes  the  nodal  transverse 
displacements  and  its  derivatives.  Through  the  use  of  Hamilton’s  principle,  the 
equations  of  motion  for  a  plate  element  undergoing  large  amplitude  vibration  may 
be  written  in  the  form 


[hb] 

0 


T  [H] 

[kB]l 

r[klNm]+[klNB]  [klbml 

,r[k2b]  < 

[k»]J 

L  ^'"*1  0  J' 

[  0  ( 

or 

[m]{  w}  +  m  +  [kl]  +  [k2]  }{w}  =  0  (5) 


where  [m]  and  [k]  are  constant  matrices  representing  the  element  mass  and  linear 
stiffiaess  characteristics,  respectively;  Pel]  and  Pe2]  are  the  first  order  and  second 
order  non-linear  sfiffiiess  matrices,  respectively;  PcInJ  depends  linearly  on 
unknown  membrane  displacement  ({Nm  }=  [A][Bni]{wni});  PcInb]  depends  linearly 
on  the  unknown  transverse  displacement  ({NB}=[B][Bb]{Wb});  DelbnJ  depends 
linearly  on  the  unknown  plate  slopes  and  represents  coupling  between  membrane 
and  bending  displacements;  and  [k2b]  depends  quadratically  on  the  unknown  plate 
slopes. 


System  Equations 

After  assembling  the  individual  finite  elements  for  the  complete  plate  and 
applying  the  kinematic  boundary  conditions,  the  finite  element  system  equations  of 
motion  for  the  large-amplitude  free  vibration  of  a  thin  laminated  composite  plate 
can  be  expressed  as 

[M]{w}4-  ([K]+  [K1(W)]  +  [K2(W)]){W}  =  0  (6) 

where  [M]  and  [K]  are  constant  matrices  and  represent  the  system  mass  and 
stifiBness  respectively;  and  [Kl]  and  [K2]  are  the  first  and  second  order  nonlinear 
stifi&iess  matrices  and  depend  linearly  and  quadratically  on  the  unknown  structural 


1377 


nodal  displacements  { W},  respectively.  Most  of  the  finite  element  large  amplitude 
fi-ee  vibration  results  for  plates  and  beams  in  the  literature,  e.g.  references  [1,6]  and 
others,  were  based  on  eq.  (6)  using  an  iterative  scheme  and  various  approximate 
procedures.  The  system  equations  are  not  suitable  for  direct  numerical  integration 
because:  a)  the  nonlinear  stif&iess  matrices  [Kl]  and  [K2]  are  functions  of  the 
unknown  nodal  displacements,  and  b)  the  number  of  degrees  of  fireedom  (DOF)  of 
the  system  nodal  displacements  {W}  is  usually  too  large.  Therefore,  eq.  (6)  has  to 
be  transformed  into  modal  or  generalized  coordinates  followed  by  a  reduction  of 
the  number  of  DOF.  In  addition,  the  general  DufiSng-type  modal  equations  will 
have  constant  nonlinear  modal  stifihiess  matrices.  This  is  accomplished  by  a  modal 
transformation  and  truncation 


r=l 

where  and  cou  are  the  natural  mode  (normalized  with  the  maximum 

component  to  unity)  and  linear  firequency  from  the  eigen-solution  ©|_j.[M]{ 
(|))«=[K]{  (|)}®. 

The  nonlinear  stififtiess  matrices  [Kl]  and  [K2]  in  eq.  (6)  can  now  be  expressed 
as  the  sum  of  the  products  of  modal  coordinates  and  nonlinear  modal  stiffaess 
matrices  as 


[Kl]  =  ^q,(t)[Kl((])«)]  (8) 

r=I 

n  n 

[K2]  =  ^  ^  qr  (t)qs  (t)[K2((|)('') ,  )]  (9) 

r=l  s=l 

The  nonlinear  modal  stifihiess  matrices  [Kl]^"^  and  [O]^'^^  are  assembled  from  the 
element  nonlinear  modal  stif&iess  terms  [kl]^^^  and  Pc2]^“^  as 

([K1]«.[K2]‘“))=  ^([klf>,[k2]'“>)  (10) 

al!  elements 
+  bdy.  conds. 


where  the  element  nonlinear  modal  stif&iess  matrices  are  evaluated  with  the  known 
linear  mode  Thus,  the  nonlinear  modal  stiffiiess  [Kl]^^^  and  [K2]^'®^  are  constant 
matrices.  Equation  (6)  is  thus  transformed  to  the  general  Duf6ng-type  modal 
equations  as 


1378 


(11) 


[M]{q}+([K]+[KlJ+[K2„]){q}  =  0 
where  the  modal  mass  and  linear  stifhiess  matrices  are  diagonal 
([M],[K])  =  [<I.f([M],[K])[<I.] 
and  the  quadratic  and  cubic  terms  are 


LI 

[Kl,]{q}  =  [<Df  J^qjKl] 


1^) 


Vt=1 
n  n 


(rs) 


r=:l  s=l 


\Wk} 


(12) 


(13) 


(14) 


AH  modal  matrices  in  eq.  (11)  are  constant  matrices.  With  given  initial 
conditions,  the  modal  coordinate  responses  {q}  can  be  determined  from  eq.  (11) 
using  any  direct  numerical  integration  scheme  such  as  the  Runge-Kutta  or 
Newmark-P  method.  Therefore,  no  updating  of  the  vibration  modes  is  needed  [10]. 
For  periodic  plate  osciHations  have  the  same  period  T,  the  response  of  all  modal 
coordinates  should  also  have  the  same  period  T.  Since  the  initial  conditions  wHl 
affect  greatly  the  modal  response,  the  determination  of  initial  conditions  for  periodic 
plate  osciHations  is  to  relate  each  of  the  rest  modal  coordinates  in  powers  of  the 
dominated  coordinate  as 


arqi(t;IC)  +  brqi(t;IC)  +  Crq^(t;IC)+ . =  qr(t;IC),  r  =  2,3.  ...n 

(15) 

where  the  2k,  br,  Cr  ,  .  are  constants  to  be  determined,  and  IC  denotes  initial 

conditions.  For  a  three-mode  (n=3)  system,  it  is  accurate  enough  to  keep  up  to  the 
cubic  term  only  in  eq.  (15)  and  this  leads  to  two  set  of  equations 

a2qi(tp;A,B,C)  +  b2qf(tp;A,B,C)  +  C2qJ(tp;A,B,C)  =  q2(tp;A,B,C),  p  =  1,2,3 
a3qi  (tp ; A,  B, C)  +  bsq J  (t p ;  A,  B.  C)  +  Csq J  (tp ; A.  B,  C)  =  q3  (tp ; A,  B,  C),  p  =  1,2,3 

(16a,b) 

in  which  the  modal  coordinates  qi,  q2  and  qs  at  tp  are  known  quantities  and  the 
initial  conditions  are  qi(0)=A,  q2(0)  =B,  q3(0)=C  and  qi(0)  =  q2(0)  =  q3(0)  =  0. 
PracticaHy,  only  eight  equations  are  needed  to  determine  the  eight  unknowns  a2,  as. 


1379 


hi,  b3,  C2,  C3,  B  and  C  through  an  iterative  scheme.  However,  the  number  of 
equations  can  be  more  than  the  number  of  unknowns  for  accurate  determination  of 
initial  conditions  and  the  least  square  method  is  employed  in  this  case. 

The  time  history  of  the  plate  maximum  deflection  can  be  obtained  from  eq.  (7). 
The  participation  value  from  the  r  th  linear  mode  to  the  total  deflection  is  defined 
as 


maxjqj. 

n 

^max|qi 

i=l 


(17) 


Thus,  the  minimum  number  of  the  linear  modes  for  an  accurate  and  converged 
frequency  solution  can  be  determined  based  on  the  modal  participation  values. 


Results  and  Discussions 

Assessment  of  Single-Mode  Elliptic  Function  Solution 

The  fundamental  frequency  ratio  Co/col  =  2.3051  at  Wmax/r  =5.0  for  a  simply 
supported  beam  obtained  by  Woinowsky-Kiieger  [2]  using  a  single-mode  and 
elliptic  fimetion  is  assessed  first.  The  conventional  beam  element  having  six  (four 
bending  and  two  axial)  DOF  is  used.  A  half-beam  is  modeled  with  10,  15,  20 
elements,  and  the  lowest  four  symmetrical  linear  modes  are  used  in  the  Duffing 
modal  equations.  Table  1  shows  that  a  20-element  and  1-mode  model  will  yield  a 
converged  result.  The  percentages  of  participation  from  each  mode  for  various 
values  of  Wmax/r  are  given  in  Table  2.  The  modal  participation  values  demonstrate 
that  a  single  mode  (n=l)  will  yield  an  accurate  fundamental  frequency  because  the 
contribution  from  higher  linear  modes  to  the  total  deflection  is  negligible  (<  0.01  % 
for  Wmax/r  up  to  5.0).  There  is  a  small  difference  in  frequency  ratios  between  the 
present  finite  element  and  the  elliptic  integral  solutions.  This  is  due  to  the  difference 
between  the  axial  forces  of  the  two  approaches,  the  finite  element  method  (FEM) 
gives  a  non-uniform  axial  force  in  each  element;  however,  the  average  value  of  the 
axial  force  for  each  element  is  the  same  as  the  one  in  the  classic  continuum 
approach.  The  lowest  (2.0310)  and  the  highest  (3.3438)  frequency  ratios  at  Wmax/r 
=5.0  in  reference  [5]  are  not  accurate. 

Frequency  ratios  for  higher  modes  of  the  simply  supported  beam  are  obtained 
next.  A  model  with  40-elements  and  3-anti-symmetric  modes  for  the  whole  beam  is 
employed  for  the  frequency  ratio  of  the  second  nonlinear  mode.  The  mode 
participations  shown  in  Table  2  indicate  that  a  single-mode  approach  will  yield 
accurate  frequency  results.  And  the  frequency  ratios  for  the  second  mode  are  the 
same  as  those  of  the  fundamental  one.  Thus,  the  present  method  agrees  extremely 
well  with  Woinowsky-Krieger’s  classic  single-mode  approach. 


1380 


The  time  history  of  the  first  two  symmetric  modal  coordinates  and  the  beam 
central  displacement,  phase  plot,  and  power  spectral  density  (PSD)  at  maximum 
beam  deflection  W^Jr  =  5.0  for  the  fimdamental  firequency  (or  mode)  are  shown  in 
Fig.  1.  The  time  scale  is  non-dimensional  and  Ti  is  the  period  of  the  fundamental 
linear  resonance.  It  is  noted  that  although  the  central  displacement  response  looks 
like  a  simple  harmonic  motion,  it  does  have  a  small  deviation  fi:om  pure  harmonic 
motion  due  to  the  second  small  peak  in  the  spectrum.  This  is  in  agreement  with 
classical  solution  that  the  ratio  of  the  jfrequency  of  the  second  small  peak  to  that  of 
the  first  dominant  peak  is  3. 

Now  we  are  ready  to  assess  the  single-mode  fundamental  firequency  of  a  simply 
supported  square  plate  obtained  by  Chu  and  Herrmann  [3].  A  quarter  of  the  plate  is 
modeled  with  6  x  6,  7  x  7,  8  x  8  and  9  x  9  mesh  sizes  and  1,  2,  4  or  5  symmetrical 
modes  are  used.  The  conforming  rectangular  plate  element  with  24  (16  bending 
and  8  membrane )  DOF  is  used.  The  in-plane  boundary  conditions  are  u  =  v  =  0  on 
all  four  edges.  Table  3  shows  that  the  8  x  8  mesh  size  in  a  quarter-plate  and  4- 
mode  model  should  be  used  for  a  converged  and  accurate  frequency  solution.  Table 
4  shows  the  fi:equency  ratios  and  modal  participation  values  for  the  lowest  three 
modes  at  various  Wmax/h  for  a  simply  supported  square  plate  (8x8  mesh  size  in  a 
quarter-plate).  It  indicated  that  at  least  two  linear  modes  are  needed  for  an  accurate 
frequency  prediction  at  Wmax/h  =1.0,  and  the  contribution  of  higher  linear  modes 
increase  with  the  increase  of  plate  deflections.  The  modal  participation  values  also 
show  that  the  combined  modes  (1,3)-(3,1)  and  (2,4)-(4,2)  are  independent  of  the 
large-amplitude  vibrations  dominated  by  (1,1)  and  (2,2)  modes,  respectively.  The 
time  history,  phase  plot,  and  PSD  at  the  maximum  deflection  Wmax/h  =1.0  for  the 
fundamental  mode  are  shown  in  Fig.  2a,  and  Tu  is  the  period  of  the  fundamental 
linear  resonance.  There  is  one  small  peak  in  the  spectrum  and  the  frequency  ratio  of 
the  second  small  peak  to  the  first  dominant  one  is  3.  The  low  (1.2967)  and  the  high 
(1.5314)  frequency  ratios  at  Wmax/h  =1.0  given  in  reference  [6]  are  not  accurate. 

The  influence  of  the  initial  conditions  on  periodic  motion  is  demonstrated  in  Fig. 
2a  and  2b.  In  Fig.  2a,  the  modal  coordinates  all  have  the  same  period,  and  the  initial 
conditions  are  determined  fi*om  eq.(15).  They  are  qii(0)/h=1.0,  qi3+3i(0)/h=  - 
0.0155,  qi3.3i  (0)/h  =  0.0,  q33(0)/h=0.000813,  and  qi5+5i(0)/h=  0.00011,  and  initial 
velocities  are  null,  whereas  in  Fig.  2b,  qii(0)/h=1.0  and  all  others  are  nuU.  The 
modal  coordinates  do  not  have  the  same  period. 

Clamped  Beam 

It  is  thus  curious  to  find  out  whether  multiple-mode  is  required  for  the  clamped 
beam.  Convergence  study  of  the  fundamental  firequency  ratios  at  Wmax/r  =3.0  and 
5.0  shown  in  Table  5  indicates  that  a  25-element  (half-beam)  and  4-mode  model 
win  yield  accurate  and  converged  results.  The  time  history,  phase  plot  and  PSD  at 
Wmax/r  =5.0  are  shown  in  Fig.  3.  The  modal  participation  values  in  Table  6  and  the 
PSD  in  Fig.  3  confirm  that  at  least  two  modes  are  needed  for  accurate  firequency 
results. 


1381 


Symmetric  Composite  Plate 

A  simply  supported  eight-layer  symmetrically  laminated  (0/45/-45/90)s 
composite  plate  with  an  aspect  ratio  of  2  is  investigated.  The  graphite/epoxy 
material  properties  are  as  follows;  Young’s  moduli  Ei  =  155  GPa,  E2  =  8.07  GPa, 
shear  modulus  Gn  =  4.55GPa,  Poisson’s  ratio  V12  =0.22,  and  mass  density  p  = 
1550  kg/m^  A  12  X  12  mesh  is  used  to  model  the  plate.  The  in-plane  boundary 
conditions  are  fixed  (u=v=0)  at  all  four  edges.  The  first  seven  linear  modes  are  used 
as  the  modal  coordinates.  Table  7  gives  the  fundamental  firequency  ratios  and  mode 
participation  values  for  the  linear  modes  in  increasing  firequency  order.  The  modal 
participation  values  indicate  clearly  that  four  modes  are  needed  in  predicting  the 
nonlinear  fi-equency,  and  other  three  of  the  seven  are  independent  of  the 
fundamental  nonlinear  mode.  Figure  4  shows  the  time-history,  phase  plot,  and  PSD 
at  Wmax/h  =1.0. 


UNS  YMMETRIC  COMPOSITE  PLATE 

A  simply  supported  two-layer  laminated  (0/90)  composite  plate  of  15  x  12  x 
0.048  in.  (38  x  30  x  0.12  cm)  is  investigated.  The  graphite/epoxy  material 
properties  are  the  same  as  those  of  the  symmetric  composite  plate.  A 12  x  12  mesh 
is  used  to  model  the  plate.  The  in-plane  boundary  conditions  are  fixed  at  all  four 
edges.  The  first  four  linear  modes  are  used  as  the  modal  coordinates.  Table  8  gives 
the  fundamental  firequency  ratios  and  mode  participation  values  for  the  linear  modes 
in  increasing  fi-equency  order.  From  the  phase  plot,  the  time  histories  and  PSD 
shown  in  Fig.  5,  it  can  be  seen  that  the  total  displacement  response  has  a  non-zero 
mean  (i.e.  the  positive  and  negative  displacement  amplitudes  for  all  modal 
coordinates  are  not  equal).  The  quasi-ellipse  in  the  phase  plot  is  not  symmetrical 
about  the  vertical  velocity-axis.  In  the  PSD  at  Wmax/h  =1.0,  it  is  observed  that  there 
are  four  small  peaks  in  the  spectrum  and  the  firequency  ratios  of  the  second,  third, 
fourth  and  fifth  peak  to  the  first  dominant  one  are  2,  3,  4  and  5,  respectively.  This 
observation  indicates  that  the  displacement  response  includes  the 
superharmonances  of  orders  2,  3,  4,  and  5.  The  curves,  which  the  positive  and 
negative  displacement  amplitudes  are  plotted  against  the  fundamental  firequency 
ratio,  are  also  given  in  Fig.  5.  The  difference  between  the  positive  and  negative 
amplitudes  increases  as  the  firequency  ratio  increasing. 


Conclusions 

A  multimode  time-domain  formulation,  based  on  the  finite  element  method,  is 
presented  for  nonlinear  firee  vibration  of  composite  plates.  The  use  of  FEM  enables 
the  present  formulation  to  deal  with  composite  plates  of  complex  geometries  and 
boundary  conditions,  and  the  use  of  the  modal  coordinate  transformation  enables  to 
reduce  the  number  of  ordinary  nonlinear  differential  modal  equations  to  a  much 
smaller  one.  The  present  procedure  is  able  to  obtain  the  general  Duffing-type  modal 
equations  easily.  Initial  conditions  for  all  modal  coordinates  having  the  same  time 


1382 


period  are  presented.  The  participation  value  of  the  linear  mode  to  the  nonlinear 
deflection  is  quantified  ;  they  can  clearly  determine  the  minimum  number  of  linear 
modes  needed  for  accurate  nonlinear  frequency  results. 

The  present  fundamental  nonlinear  fi-equency  ratios  have  been  compared  with 
the  single-mode  solution  obtained  by  Woinowsky-Kneger  for  simply  supported 
beams  and  by  Chu  and  Herrmann  for  simply  supported  square  plates.  The 
Woinowsky-Krieger’s  single-mode  solution  is  accurate.  For  all  other  solutions, 
however,  two  or  more  modes  are  needed.  The  nonlinear  firequencies  for 
symmetrically  and  unsymmetrically  laminated  rectangular  composite  plates  are  also 
obtained.  The  phase  plot  and  power  spectral  density  showed  that  nonlinear 
displacement  responses  are  no  longer  harmonic,  and  multiple  modes  are  required 
for  isotropic  clamped  beams  and  isotropic  and  composite  plates. 


References 

1.  M.  Sathyamoorthy  1987  Applied  Mechanics  Review  40,  1553-1561.  Nonlinear 
vibration  analysis  of  plates:  A  review  and  survey  of  current  developments. 

2.  S.  Woinowsky-Kreger  1950  Journal  of  Applied  Mechanics  17,  35-36.  The  effect  of  an 
axial  force  on  the  vibration  of  hinged  bars. 

3.  H.  N.  Chu  and  G.  Herrmann  1956  Journal  of  Applied  Mechanics  23,  523-540. 
Influence  of  large  amplitudes  on  jfree  flexural  vibrations  of  rectangular  elastic  plates. 

4.  G.  Singh,  A.  k.  Sharma  and  G.  V.  Rao  1990  Journal  of  Sound  and  Vibration  142,  77- 
85.  Large  amplitude  free  vibration  of  beams-discussion  of  various  formulations  and 
assumptions. 

5.  H.  R.  Srirangaraja  1994  Journal  of  Sound  and  Vibration  175,  425-427.  Nonlinear  free 
vibrations  of  uniform  beams. 

6.  S.  R.  Rao,  A.  H.  Sheikh  and  M.  Mukhopadhyay  1993  Journal  of  the  Acoustical 
Society  of  America  93  (6),  3250-3257.  Large-amplitude  finite  element  flexural 
vibration  of  plates/stiffened  plates. 

7.  Y.  Shi  and  C.  Mei  1996  Journal  Sound  and  Vibration  193,  453-464.  A  finite  element 
time  domain  modal  formulation  for  large  amplitude  free  vibration  of  beams  and  plates. 

8.  K.  Bogna:,  R.  L.  Fox  and  L.  A.  Schmit  1966  Proceeding  of  Conference  on  Matrix 
Methods  in  Structural  Mechanics,  AFFDL-TR-66-80,  Wright-Patterson  Air  Force 
Base,  Ohio,  October  1965,  397-444.  The  gena-ation  of  interelement  compatible 
stiffness  and  mass  matrix  using  the  interpolation  formulas. 

9.  Teseller  and  T.  J.  R.  Hughes  1985  Computer  Methods  in  Applied  Mechanics  and 
Engineering  50,  71  -101 .  A  three  node  Mindlin  plate  element  with  improved  fransverse 
shear. 

10.  A.  K.  Noor  1981  Composites  and  Structures,  13,  31-44.  Recent  advances  in  reduction 
methods  for  nonlinear  problems. 


1383 


Table  1 .  Convergence  of  the  fundamental  frequency  ratio  at  Wmax/r  =5.0  for  a 
simply  supported  beam 


No.  of  elements 
and  4  modes 

(C0/(0l)i 

No.  of  modes 
and  20  elements 

(G)/G)l)i 

10 

2.3537 

1 

2.3506 

15 

2.3511 

2 

2.3506 

20 

2.3506 

3 

2.3506 

-- 

4 

2.3506 

Table  2  The  lowest  two  frequency  ratios  and  the  modal  participations  for  a 
simply  supported  beam 


W^nax/r 

Elliptic 
integral  [2] 
(CO/OOl)! 

FEM 

Modal 

Participation  % 

(CO/COl)! 

_ _ 

0.2 

1.0038 

1.0038 

100.00 

0.000 

0.000 

0.4 

1.0150 

1.0149 

100.00 

0.000 

0.000 

0.6 

1.0331 

1.0331 

100.00 

0.000 

0.000 

0.8 

1.0580 

1.0581 

100.00 

0.000 

0.000 

1 

1.0892 

1.0892 

100.00 

0.000 

0.000 

2 

1.3178 

1.3179 

100.00 

0.002 

0.000 

3 

1.6257 

1.6258 

100.00 

0.004 

0.000 

4 

1.9760 

1.9761 

99.99 

0.005 

0.000 

5 

2.3501 

2.3506 

99.99 

0.009 

0.000 

W.„ax/r 

(co/g)l)2 

(C0/CDl)2 

q.2_  _ 

q4 

Q6 

0.2 

1.0038 

1.0038 

100.00 

0.000 

0.000 

0.4 

1.0150 

1.0149 

100.00 

0.000 

0.000 

0.6 

1.0331 

1.0332 

100.00 

0.000 

0.001 

0.8 

1.0580 

1.0582 

100.00 

0.000 

0.001 

1 

1.0892 

1.0893 

100.00 

0.000 

0.002 

2 

1.3178 

1.3181 

99.99 

0.000 

0.006 

3 

1.6257 

1.6260 

99.98 

0.000 

0.015 

4 

1.9760 

1.9768 

99.98 

0.000 

0.021 

5 

2.3501 

2.3512 

99.96 

0.000 

0.037 

Table  3.  Convergence  of  the  fundamental  frequency  ratios  for  a  simply 
supported  square  plate  (Poisson’s  ratio=0.3) 


Mesh  sizes 
and 

(CD/Qjn 

at 

W^,Jh 

No.  of  modes 
and 

(co/coUn 

at 

W^Jh 

4  modes 

1.0 

1.4 

8x8  mesh 

1.0 

1.4 

6x6 

1.4174 

1.7423 

1 

1.7028 

7x7 

1.4163 

1.7396 

2 

1.4169 

1,7433 

8x8 

1.4164 

1.7403 

4 

1.4164 

1.7403 

9x9 

1.4164 

1.7400 

5 

1.4163 

1.7401 

Table  4.  The  lowest  three  frequency  ratios  and  the  modal  participations  for  a 
simply  supported  square  plate  (Poisson’s  ratio=0.3) 


Elliptic 

FEM 

integral [3] 

(CO/ffiiJii 

(to/oiOii 

Modal 

Participation 

% 

qu 

qi3  -1-  qsi 

qi3  -  qsi 

q33 

qi5  -f.  qsi 

0.2 

1.0195 

1.0195 

99.93 

0.07 

0.00 

0.00 

0.00 

0.4 

1.0757 

1.0765 

99.72 

0.27 

0.00 

0.01 

0.00 

0.6 

1.1625 

1.1658 

99.38 

0.59 

0.00 

0.02 

0.00 

0.8 

1.2734 

1.2796 

98.93 

1.02 

0.00 

0.05 

0.01 

1.0 

1.4024 

1.4163 

98.34 

1.57 

0.00 

0.08 

0.01 

1.2 

1.5448 

1.5659 

97.54 

2.30 

0.00 

0.15 

0.01 

1.4 

1,6933 

1.7401 

96.29 

3.42 

0.00 

0.27 

0.02 

(C0/®l)21 

q2i 

q23 

q4i 

q43 

... 

0.2 

N/A 

1.0243 

99.93 

0.06 

0.01 

0.00 

... 

0.4 

N/A 

1.0976 

99.50 

0.45 

0.03 

0.01 

— 

0.6 

N/A 

1.2072 

98.15 

1.28 

0.54 

0.02 

... 

0.8 

N/A 

1.3411 

97.54 

2.41 

0.00 

0.05 

— 

1.0 

N/A 

1.5126 

96.24 

3.69 

0.00 

0.08 

... 

1.2 

N/A 

1.6900 

94.90 

4.92 

0.03 

0.15 

... 

1.4 

N/A 

1.8952 

93.54 

6.01 

0.01 

0.44 

... 

y/raJh 

(C0/0)l)22 

q22 

q24  +  q42 

q24  -  q42 

q44 

... 

0.2 

N/A 

1.0245 

100.00 

0.00 

0.00 

... 

0.4 

N/A 

1.0751 

100.00 

0.00 

0.00 

0.00 

... 

0.6 

N/A 

1.1611 

99.99 

0.00 

0.00 

0.01 

... 

0.8 

N/A 

1.2806 

99.98 

0.01 

0.00 

0.01 

... 

1.0 

N/A 

1.4041 

99.93 

0.01 

0.00 

0.06 

— 

1.2 

N/A 

1.5551 

99.97 

0.01 

0.00 

0.0